fix(curriculum): clean-up Project Euler 441-460 (#43068)
* fix: clean-up Project Euler 441-460 * fix: corrections from review Co-authored-by: Tom <20648924+moT01@users.noreply.github.com> Co-authored-by: Tom <20648924+moT01@users.noreply.github.com>
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@ -8,20 +8,24 @@ dashedName: problem-441-the-inverse-summation-of-coprime-couples
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# --description--
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For an integer M, we define R(M) as the sum of 1/(p·q) for all the integer pairs p and q which satisfy all of these conditions:
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For an integer $M$, we define $R(M)$ as the sum of $\frac{1}{p·q}$ for all the integer pairs $p$ and $q$ which satisfy all of these conditions:
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1 ≤ p < q ≤ M p + q ≥ M p and q are coprime.
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- $1 ≤ p < q ≤ M$
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- $p + q ≥ M$
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- $p$ and $q$ are coprime.
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We also define S(N) as the sum of R(i) for 2 ≤ i ≤ N. We can verify that S(2) = R(2) = 1/2, S(10) ≈ 6.9147 and S(100) ≈ 58.2962.
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We also define $S(N)$ as the sum of $R(i)$ for $2 ≤ i ≤ N$.
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Find S(107). Give your answer rounded to four decimal places.
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We can verify that $S(2) = R(2) = \frac{1}{2}$, $S(10) ≈ 6.9147$ and $S(100) ≈ 58.2962$.
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Find $S({10}^7)$. Give your answer rounded to four decimal places.
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# --hints--
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`euler441()` should return 5000088.8395.
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`inverseSummationCoprimeCouples()` should return `5000088.8395`.
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```js
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assert.strictEqual(euler441(), 5000088.8395);
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assert.strictEqual(inverseSummationCoprimeCouples(), 5000088.8395);
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```
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# --seed--
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@ -29,12 +33,12 @@ assert.strictEqual(euler441(), 5000088.8395);
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## --seed-contents--
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```js
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function euler441() {
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function inverseSummationCoprimeCouples() {
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return true;
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}
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euler441();
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inverseSummationCoprimeCouples();
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```
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# --solutions--
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@ -12,16 +12,16 @@ An integer is called eleven-free if its decimal expansion does not contain any s
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For example, 2404 and 13431 are eleven-free, while 911 and 4121331 are not.
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Let E(n) be the nth positive eleven-free integer. For example, E(3) = 3, E(200) = 213 and E(500 000) = 531563.
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Let $E(n)$ be the $n$th positive eleven-free integer. For example, $E(3) = 3$, $E(200) = 213$ and $E(500\\,000) = 531\\,563$.
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Find E(1018).
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Find $E({10}^{18})$.
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# --hints--
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`euler442()` should return 1295552661530920200.
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`elevenFreeIntegers()` should return `1295552661530920200`.
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```js
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assert.strictEqual(euler442(), 1295552661530920200);
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assert.strictEqual(elevenFreeIntegers(), 1295552661530920200);
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```
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# --seed--
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@ -29,12 +29,12 @@ assert.strictEqual(euler442(), 1295552661530920200);
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## --seed-contents--
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```js
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function euler442() {
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function elevenFreeIntegers() {
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return true;
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}
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euler442();
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elevenFreeIntegers();
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```
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# --solutions--
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@ -8,24 +8,30 @@ dashedName: problem-443-gcd-sequence
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# --description--
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Let g(n) be a sequence defined as follows: g(4) = 13, g(n) = g(n-1) + gcd(n, g(n-1)) for n > 4.
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Let $g(n)$ be a sequence defined as follows:
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$$\begin{align}
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& g(4) = 13, \\\\
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& g(n) = g(n-1) + gcd(n, g(n - 1)) \text{ for } n > 4.
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\end{align}$$
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The first few values are:
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n 4567891011121314151617181920... g(n) 1314161718272829303132333451545560...
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$$\begin{array}{l}
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n & 4 & 5 & 6 & 7 & 8 & 9 & 10 & 11 & 12 & 13 & 14 & 15 & 16 & 17 & 18 & 19 & 20 & \ldots \\\\
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g(n) & 13 & 14 & 16 & 17 & 18 & 27 & 28 & 29 & 30 & 31 & 32 & 33 & 34 & 51 & 54 & 55 & 60 & \ldots
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\end{array}$$
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<!-- TODO Use MathJax -->
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You are given that $g(1\\,000) = 2\\,524$ and $g(1\\,000\\,000) = 2\\,624\\,152$.
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You are given that g(1 000) = 2524 and g(1 000 000) = 2624152.
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Find g(1015).
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Find $g({10}^{15})$.
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# --hints--
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`euler443()` should return 2744233049300770.
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`gcdSequence()` should return `2744233049300770`.
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```js
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assert.strictEqual(euler443(), 2744233049300770);
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assert.strictEqual(gcdSequence(), 2744233049300770);
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```
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# --seed--
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@ -33,12 +39,12 @@ assert.strictEqual(euler443(), 2744233049300770);
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## --seed-contents--
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```js
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function euler443() {
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function gcdSequence() {
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return true;
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}
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euler443();
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gcdSequence();
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```
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# --solutions--
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@ -8,7 +8,7 @@ dashedName: problem-444-the-roundtable-lottery
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# --description--
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A group of p people decide to sit down at a round table and play a lottery-ticket trading game. Each person starts off with a randomly-assigned, unscratched lottery ticket. Each ticket, when scratched, reveals a whole-pound prize ranging anywhere from £1 to £p, with no two tickets alike. The goal of the game is for each person to maximize his ticket winnings upon leaving the game.
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A group of $p$ people decide to sit down at a round table and play a lottery-ticket trading game. Each person starts off with a randomly-assigned, unscratched lottery ticket. Each ticket, when scratched, reveals a whole-pound prize ranging anywhere from £1 to £$p$, with no two tickets alike. The goal of the game is for each person to maximize his ticket winnings upon leaving the game.
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An arbitrary person is chosen to be the first player. Going around the table, each player has only one of two options:
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@ -19,18 +19,26 @@ The game ends once all tickets have been scratched. All players still remaining
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Assume that each player uses the optimal strategy for maximizing the expected value of his ticket winnings.
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Let E(p) represent the expected number of players left at the table when the game ends in a game consisting of p players (e.g. E(111) = 5.2912 when rounded to 5 significant digits).
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Let $E(p)$ represent the expected number of players left at the table when the game ends in a game consisting of $p$ players (e.g. $E(111) = 5.2912$ when rounded to 5 significant digits).
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Let S1(N) = E(p) Let Sk(N) = Sk-1(p) for k > 1
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Let $S_1(N) = \displaystyle\sum_{p = 1}^N E(p)$.
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Find S20(1014) and write the answer in scientific notation rounded to 10 significant digits. Use a lowercase e to separate mantissa and exponent (e.g. S3(100) = 5.983679014e5).
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Let $S_k(N) = \displaystyle\sum_{p = 1}^N S_{k - 1}(p)$ for $k > 1$.
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Find $S_{20}({10}^{14})$ and write the answer as a string in scientific notation rounded to 10 significant digits. Use a lowercase `e` to separate mantissa and exponent. For example, the answer for $S_3(100)$ would be `5.983679014e5`.
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# --hints--
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`euler444()` should return 1.200856722e+263.
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`roundtableLottery()` should return a string.
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```js
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assert.strictEqual(euler444(), 1.200856722e263);
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assert(typeof roundtableLottery() === 'string');
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```
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`roundtableLottery()` should return the string `1.200856722e263`.
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```js
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assert.strictEqual(roundtableLottery(), '1.200856722e263');
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```
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# --seed--
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@ -38,12 +46,12 @@ assert.strictEqual(euler444(), 1.200856722e263);
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## --seed-contents--
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```js
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function euler444() {
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function roundtableLottery() {
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return true;
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}
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euler444();
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roundtableLottery();
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```
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# --solutions--
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@ -8,20 +8,26 @@ dashedName: problem-445-retractions-a
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# --description--
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For every integer n>1, the family of functions fn,a,b is defined
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For every integer $n > 1$, the family of functions $f_{n, a, b}$ is defined by:
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by fn,a,b(x)≡ax+b mod n for a,b,x integer and 0
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$f_{n, a, b}(x) ≡ ax + b\bmod n$ for $a, b, x$ integer and $0 \lt a \lt n$, $0 \le b \lt n$, $0 \le x \lt n$.
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You are given that ∑ R(c) for c=C(100 000,k), and 1 ≤ k ≤99 999 ≡628701600 (mod 1 000 000 007). (C(n,k) is the binomial coefficient).
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We will call $f_{n, a, b}$ a retraction if $f_{n, a, b}(f_{n, a, b}(x)) \equiv f_{n, a, b}(x)\bmod n$ for every $0 \le x \lt n$.
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Find ∑ R(c) for c=C(10 000 000,k), and 1 ≤k≤ 9 999 999. Give your answer modulo 1 000 000 007.
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Let $R(n)$ be the number of retractions for $n$.
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You are given that
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$$\sum_{k = 1}^{99\\,999} R(\displaystyle\binom{100\\,000}{k}) \equiv 628\\,701\\,600\bmod 1\\,000\\,000\\,007$$
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Find $$\sum_{k = 1}^{9\\,999\\,999} R(\displaystyle\binom{10\\,000\\,000}{k})$$ Give your answer modulo $1\\,000\\,000\\,007$.
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# --hints--
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`euler445()` should return 659104042.
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`retractionsA()` should return `659104042`.
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```js
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assert.strictEqual(euler445(), 659104042);
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assert.strictEqual(retractionsA(), 659104042);
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```
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# --seed--
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@ -29,12 +35,12 @@ assert.strictEqual(euler445(), 659104042);
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## --seed-contents--
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```js
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function euler445() {
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function retractionsA() {
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return true;
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}
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euler445();
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retractionsA();
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```
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# --solutions--
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@ -8,20 +8,26 @@ dashedName: problem-446-retractions-b
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# --description--
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For every integer n>1, the family of functions fn,a,b is defined
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For every integer $n > 1$, the family of functions $f_{n, a, b}$ is defined by:
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by fn,a,b(x)≡ax+b mod n for a,b,x integer and 0
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$f_{n, a, b}(x) ≡ ax + b\bmod n$ for $a, b, x$ integer and $0 \lt a \lt n$, $0 \le b \lt n$, $0 \le x \lt n$.
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F(N)=∑R(n4+4) for 1≤n≤N. F(1024)=77532377300600.
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We will call $f_{n, a, b}$ a retraction if $f_{n, a, b}(f_{n, a, b}(x)) \equiv f_{n, a, b}(x)\bmod n$ for every $0 \le x \lt n$.
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Find F(107) (mod 1 000 000 007)
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Let $R(n)$ be the number of retractions for $n$.
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$F(N) = \displaystyle\sum_{n = 1}^N R(n^4 + 4)$.
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$F(1024) = 77\\,532\\,377\\,300\\,600$.
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Find $F({10}^7)$. Give your answer modulo $1\\,000\\,000\\,007$.
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# --hints--
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`euler446()` should return 907803852.
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`retractionsB()` should return `907803852`.
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```js
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assert.strictEqual(euler446(), 907803852);
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assert.strictEqual(retractionsB(), 907803852);
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```
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# --seed--
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@ -29,12 +35,12 @@ assert.strictEqual(euler446(), 907803852);
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## --seed-contents--
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```js
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function euler446() {
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function retractionsB() {
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return true;
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}
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euler446();
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retractionsB();
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```
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# --solutions--
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@ -8,20 +8,26 @@ dashedName: problem-447-retractions-c
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# --description--
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For every integer n>1, the family of functions fn,a,b is defined
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For every integer $n > 1$, the family of functions $f_{n, a, b}$ is defined by:
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by fn,a,b(x)≡ax+b mod n for a,b,x integer and 0
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$f_{n, a, b}(x) ≡ ax + b\bmod n$ for $a, b, x$ integer and $0 \lt a \lt n$, $0 \le b \lt n$, $0 \le x \lt n$.
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F(N)=∑R(n) for 2≤n≤N. F(107)≡638042271 (mod 1 000 000 007).
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We will call $f_{n, a, b}$ a retraction if $f_{n, a, b}(f_{n, a, b}(x)) \equiv f_{n, a, b}(x)\bmod n$ for every $0 \le x \lt n$.
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Find F(1014) (mod 1 000 000 007).
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Let $R(n)$ be the number of retractions for $n$.
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$F(N) = \displaystyle\sum_{n = 2}^N R(n)$.
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$F({10}^7) ≡ 638\\,042\\,271\bmod 1\\,000\\,000\\,007$.
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Find $F({10}^{14})$. Give your answer modulo $1\\,000\\,000\\,007$.
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# --hints--
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`euler447()` should return 530553372.
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`retractionsC()` should return `530553372`.
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```js
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assert.strictEqual(euler447(), 530553372);
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assert.strictEqual(retractionsC(), 530553372);
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```
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# --seed--
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@ -29,12 +35,12 @@ assert.strictEqual(euler447(), 530553372);
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## --seed-contents--
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```js
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function euler447() {
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function retractionsC() {
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return true;
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}
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euler447();
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retractionsC();
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```
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# --solutions--
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@ -8,22 +8,24 @@ dashedName: problem-448-average-least-common-multiple
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# --description--
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The function lcm(a,b) denotes the least common multiple of a and b.
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The function $lcm(a, b)$ denotes the least common multiple of $a$ and $b$.
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Let A(n) be the average of the values of lcm(n,i) for 1≤i≤n.
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Let $A(n)$ be the average of the values of $lcm(n, i)$ for $1 ≤ i ≤ n$.
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E.g: A(2)=(2+2)/2=2 and A(10)=(10+10+30+20+10+30+70+40+90+10)/10=32.
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E.g: $A(2) = \frac{2 + 2}{2} = 2$ and $A(10) = \frac{10 + 10 + 30 + 20 + 10 + 30 + 70 + 40 + 90 + 10}{10} = 32$.
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Let S(n)=∑A(k) for 1≤k≤n. S(100)=122726.
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Let $S(n) = \sum A(k)$ for $1 ≤ k ≤ n$.
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Find S(99999999019) mod 999999017.
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$S(100) = 122\\,726$.
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Find $S(99\\,999\\,999\\,019)\bmod 999\\,999\\,017$.
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# --hints--
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`euler448()` should return 106467648.
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`averageLCM()` should return `106467648`.
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```js
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assert.strictEqual(euler448(), 106467648);
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assert.strictEqual(averageLCM(), 106467648);
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```
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# --seed--
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@ -31,12 +33,12 @@ assert.strictEqual(euler448(), 106467648);
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## --seed-contents--
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```js
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function euler448() {
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function averageLCM() {
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return true;
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}
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euler448();
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averageLCM();
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```
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# --solutions--
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@ -8,24 +8,22 @@ dashedName: problem-449-chocolate-covered-candy
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# --description--
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Phil the confectioner is making a new batch of chocolate covered candy. Each candy centre is shaped like an ellipsoid of revolution defined by the equation: b2x2 + b2y2 + a2z2 = a2b2.
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Phil the confectioner is making a new batch of chocolate covered candy. Each candy centre is shaped like an ellipsoid of revolution defined by the equation: $b^2x^2 + b^2y^2 + a^2z^2 = a^2b^2$.
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Phil wants to know how much chocolate is needed to cover one candy centre with a uniform coat of chocolate one millimeter thick. If a=1 mm and b=1 mm, the amount of chocolate required is
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Phil wants to know how much chocolate is needed to cover one candy centre with a uniform coat of chocolate one millimeter thick.
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283 π mm3
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If $a = 1$ mm and $b = 1$ mm, the amount of chocolate required is $\frac{28}{3} \pi$ mm<sup>3</sup>
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<!-- TODO Use MathJax -->
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If $a = 2$ mm and $b = 1$ mm, the amount of chocolate required is approximately 60.35475635 mm<sup>3</sup>.
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If a=2 mm and b=1 mm, the amount of chocolate required is approximately 60.35475635 mm3.
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Find the amount of chocolate in mm3 required if a=3 mm and b=1 mm. Give your answer as the number rounded to 8 decimal places behind the decimal point.
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Find the amount of chocolate in mm<sup>3</sup> required if $a = 3$ mm and $b = 1$ mm. Give your answer as the number rounded to 8 decimal places behind the decimal point.
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# --hints--
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`euler449()` should return 103.37870096.
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`chocolateCoveredCandy()` should return `103.37870096`.
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```js
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assert.strictEqual(euler449(), 103.37870096);
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assert.strictEqual(chocolateCoveredCandy(), 103.37870096);
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```
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# --seed--
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@ -33,12 +31,12 @@ assert.strictEqual(euler449(), 103.37870096);
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## --seed-contents--
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```js
|
||||
function euler449() {
|
||||
function chocolateCoveredCandy() {
|
||||
|
||||
return true;
|
||||
}
|
||||
|
||||
euler449();
|
||||
chocolateCoveredCandy();
|
||||
```
|
||||
|
||||
# --solutions--
|
||||
|
||||
@ -10,34 +10,40 @@ dashedName: problem-450-hypocycloid-and-lattice-points
|
||||
|
||||
A hypocycloid is the curve drawn by a point on a small circle rolling inside a larger circle. The parametric equations of a hypocycloid centered at the origin, and starting at the right most point is given by:
|
||||
|
||||
$x(t) = (R - r) \\cos(t) + r \\cos(\\frac {R - r} r t)$
|
||||
$$x(t) = (R - r) \cos(t) + r \cos(\frac{R - r}{r}t)$$
|
||||
|
||||
$y(t) = (R - r) \\sin(t) - r \\sin(\\frac {R - r} r t)$
|
||||
$$y(t) = (R - r) \sin(t) - r \sin(\frac{R - r}{r} t)$$
|
||||
|
||||
Where R is the radius of the large circle and r the radius of the small circle.
|
||||
Where $R$ is the radius of the large circle and $r$ the radius of the small circle.
|
||||
|
||||
Let $C(R, r)$ be the set of distinct points with integer coordinates on the hypocycloid with radius R and r and for which there is a corresponding value of t such that $\\sin(t)$ and $\\cos(t)$ are rational numbers.
|
||||
Let $C(R, r)$ be the set of distinct points with integer coordinates on the hypocycloid with radius $R$ and $r$ and for which there is a corresponding value of $t$ such that $\sin(t)$ and $\cos(t)$ are rational numbers.
|
||||
|
||||
Let $S(R, r) = \\sum\_{(x,y) \\in C(R, r)} |x| + |y|$ be the sum of the absolute values of the x and y coordinates of the points in $C(R, r)$.
|
||||
Let $S(R, r) = \sum\_{(x,y) \in C(R, r)} |x| + |y|$ be the sum of the absolute values of the $x$ and $y$ coordinates of the points in $C(R, r)$.
|
||||
|
||||
Let $T(N) = \\sum*{R = 3}^N \\sum*{r=1}^{\\lfloor \\frac {R - 1} 2 \\rfloor} S(R, r)$ be the sum of $S(R, r)$ for R and r positive integers, $R\\leq N$ and $2r < R$.
|
||||
Let $T(N) = \sum_{R = 3}^N \sum_{r=1}^{\left\lfloor \frac{R - 1}{2} \right\rfloor} S(R, r)$ be the sum of $S(R, r)$ for $R$ and $r$ positive integers, $R\leq N$ and $2r < R$.
|
||||
|
||||
You are given:C(3, 1) = {(3, 0), (-1, 2), (-1,0), (-1,-2)} C(2500, 1000) = {(2500, 0), (772, 2376), (772, -2376), (516, 1792), (516, -1792), (500, 0), (68, 504), (68, -504),(-1356, 1088), (-1356, -1088), (-1500, 1000), (-1500, -1000)}
|
||||
You are given:
|
||||
|
||||
Note: (-625, 0) is not an element of C(2500, 1000) because $\\sin(t)$ is not a rational number for the corresponding values of t.
|
||||
$$\begin{align}
|
||||
C(3, 1) = & \\{(3, 0), (-1, 2), (-1,0), (-1,-2)\\} \\\\
|
||||
C(2500, 1000) = & \\{(2500, 0), (772, 2376), (772, -2376), (516, 1792), (516, -1792), (500, 0), (68, 504), \\\\
|
||||
&(68, -504),(-1356, 1088), (-1356, -1088), (-1500, 1000), (-1500, -1000)\\}
|
||||
\end{align}$$
|
||||
|
||||
S(3, 1) = (|3| + |0|) + (|-1| + |2|) + (|-1| + |0|) + (|-1| + |-2|) = 10
|
||||
**Note:** (-625, 0) is not an element of $C(2500, 1000)$ because $\sin(t)$ is not a rational number for the corresponding values of t.
|
||||
|
||||
T(3) = 10; T(10) = 524 ;T(100) = 580442; T(103) = 583108600.
|
||||
$S(3, 1) = (|3| + |0|) + (|-1| + |2|) + (|-1| + |0|) + (|-1| + |-2|) = 10$
|
||||
|
||||
Find T(106).
|
||||
$T(3) = 10$; $T(10) = 524$; $T(100) = 580\\,442$; $T({10}^3) = 583\\,108\\,600$.
|
||||
|
||||
Find $T({10}^6)$.
|
||||
|
||||
# --hints--
|
||||
|
||||
`euler450()` should return 583333163984220900.
|
||||
`hypocycloidAndLatticePoints()` should return `583333163984220900`.
|
||||
|
||||
```js
|
||||
assert.strictEqual(euler450(), 583333163984220900);
|
||||
assert.strictEqual(hypocycloidAndLatticePoints(), 583333163984220900);
|
||||
```
|
||||
|
||||
# --seed--
|
||||
@ -45,12 +51,12 @@ assert.strictEqual(euler450(), 583333163984220900);
|
||||
## --seed-contents--
|
||||
|
||||
```js
|
||||
function euler450() {
|
||||
function hypocycloidAndLatticePoints() {
|
||||
|
||||
return true;
|
||||
}
|
||||
|
||||
euler450();
|
||||
hypocycloidAndLatticePoints();
|
||||
```
|
||||
|
||||
# --solutions--
|
||||
|
||||
@ -12,32 +12,31 @@ Consider the number 15.
|
||||
|
||||
There are eight positive numbers less than 15 which are coprime to 15: 1, 2, 4, 7, 8, 11, 13, 14.
|
||||
|
||||
The modular inverses of these numbers modulo 15 are: 1, 8, 4, 13, 2, 11, 7, 14
|
||||
The modular inverses of these numbers modulo 15 are: 1, 8, 4, 13, 2, 11, 7, 14 because
|
||||
|
||||
because
|
||||
$$\begin{align}
|
||||
& 1 \times 1\bmod 15 = 1 \\\\
|
||||
& 2 \times 8 = 16\bmod 15 = 1 \\\\
|
||||
& 4 \times 4 = 16\bmod 15 = 1 \\\\
|
||||
& 7 \times 13 = 91\bmod 15 = 1 \\\\
|
||||
& 11 \times 11 = 121\bmod 15 = 1 \\\\
|
||||
& 14 \times 14 = 196\bmod 15 = 1
|
||||
\end{align}$$
|
||||
|
||||
1\*1 mod 15=1
|
||||
Let $I(n)$ be the largest positive number $m$ smaller than $n - 1$ such that the modular inverse of $m$ modulo $n$ equals $m$ itself.
|
||||
|
||||
2\*8=16 mod 15=1
|
||||
So $I(15) = 11$.
|
||||
|
||||
4\*4=16 mod 15=1
|
||||
Also $I(100) = 51$ and $I(7) = 1$.
|
||||
|
||||
7\*13=91 mod 15=1
|
||||
|
||||
11\*11=121 mod 15=1
|
||||
|
||||
14\*14=196 mod 15=1
|
||||
|
||||
Let I(n) be the largest positive number m smaller than n-1 such that the modular inverse of m modulo n equals m itself. So I(15)=11. Also I(100)=51 and I(7)=1.
|
||||
|
||||
Find ∑I(n) for 3≤n≤2·107
|
||||
Find $\sum I(n)$ for $3 ≤ n ≤ 2 \times {10}^7$
|
||||
|
||||
# --hints--
|
||||
|
||||
`euler451()` should return 153651073760956.
|
||||
`modularInverses()` should return `153651073760956`.
|
||||
|
||||
```js
|
||||
assert.strictEqual(euler451(), 153651073760956);
|
||||
assert.strictEqual(modularInverses(), 153651073760956);
|
||||
```
|
||||
|
||||
# --seed--
|
||||
@ -45,12 +44,12 @@ assert.strictEqual(euler451(), 153651073760956);
|
||||
## --seed-contents--
|
||||
|
||||
```js
|
||||
function euler451() {
|
||||
function modularInverses() {
|
||||
|
||||
return true;
|
||||
}
|
||||
|
||||
euler451();
|
||||
modularInverses();
|
||||
```
|
||||
|
||||
# --solutions--
|
||||
|
||||
@ -8,20 +8,20 @@ dashedName: problem-452-long-products
|
||||
|
||||
# --description--
|
||||
|
||||
Define F(m,n) as the number of n-tuples of positive integers for which the product of the elements doesn't exceed m.
|
||||
Define $F(m, n)$ as the number of $n$-tuples of positive integers for which the product of the elements doesn't exceed $m$.
|
||||
|
||||
F(10, 10) = 571.
|
||||
$F(10, 10) = 571$.
|
||||
|
||||
F(106, 106) mod 1 234 567 891 = 252903833.
|
||||
$F({10}^6, {10}^6)\bmod 1\\,234\\,567\\,891 = 252\\,903\\,833$.
|
||||
|
||||
Find F(109, 109) mod 1 234 567 891.
|
||||
Find $F({10}^9, {10}^9)\bmod 1\\,234\\,567\\,891$.
|
||||
|
||||
# --hints--
|
||||
|
||||
`euler452()` should return 345558983.
|
||||
`longProducts()` should return `345558983`.
|
||||
|
||||
```js
|
||||
assert.strictEqual(euler452(), 345558983);
|
||||
assert.strictEqual(longProducts(), 345558983);
|
||||
```
|
||||
|
||||
# --seed--
|
||||
@ -29,12 +29,12 @@ assert.strictEqual(euler452(), 345558983);
|
||||
## --seed-contents--
|
||||
|
||||
```js
|
||||
function euler452() {
|
||||
function longProducts() {
|
||||
|
||||
return true;
|
||||
}
|
||||
|
||||
euler452();
|
||||
longProducts();
|
||||
```
|
||||
|
||||
# --solutions--
|
||||
|
||||
@ -10,20 +10,22 @@ dashedName: problem-453-lattice-quadrilaterals
|
||||
|
||||
A simple quadrilateral is a polygon that has four distinct vertices, has no straight angles and does not self-intersect.
|
||||
|
||||
Let Q(m, n) be the number of simple quadrilaterals whose vertices are lattice points with coordinates (x,y) satisfying 0 ≤ x ≤ m and 0 ≤ y ≤ n.
|
||||
Let $Q(m, n)$ be the number of simple quadrilaterals whose vertices are lattice points with coordinates ($x$, $y$) satisfying $0 ≤ x ≤ m$ and $0 ≤ y ≤ n$.
|
||||
|
||||
For example, Q(2, 2) = 94 as can be seen below:
|
||||
For example, $Q(2, 2) = 94$ as can be seen below:
|
||||
|
||||
It can also be verified that Q(3, 7) = 39590, Q(12, 3) = 309000 and Q(123, 45) = 70542215894646.
|
||||
<img class="img-responsive center-block" alt="94 quadrilaterals whose vertices are lattice points with coordinates (x, y) satiffying 0 ≤ x ≤ m and 0 ≤ y ≤ n" src="https://cdn.freecodecamp.org/curriculum/project-euler/lattice-quadrilaterals.png" style="background-color: white; padding: 10px;">
|
||||
|
||||
Find Q(12345, 6789) mod 135707531.
|
||||
It can also be verified that $Q(3, 7) = 39\\,590$, $Q(12, 3) = 309\\,000$ and $Q(123, 45) = 70\\,542\\,215\\,894\\,646$.
|
||||
|
||||
Find $Q(12\\,345, 6\\,789)\bmod 135\\,707\\,531$.
|
||||
|
||||
# --hints--
|
||||
|
||||
`euler453()` should return 104354107.
|
||||
`latticeQuadrilaterals()` should return `104354107`.
|
||||
|
||||
```js
|
||||
assert.strictEqual(euler453(), 104354107);
|
||||
assert.strictEqual(latticeQuadrilaterals(), 104354107);
|
||||
```
|
||||
|
||||
# --seed--
|
||||
@ -31,12 +33,12 @@ assert.strictEqual(euler453(), 104354107);
|
||||
## --seed-contents--
|
||||
|
||||
```js
|
||||
function euler453() {
|
||||
function latticeQuadrilaterals() {
|
||||
|
||||
return true;
|
||||
}
|
||||
|
||||
euler453();
|
||||
latticeQuadrilaterals();
|
||||
```
|
||||
|
||||
# --solutions--
|
||||
|
||||
@ -8,20 +8,22 @@ dashedName: problem-454-diophantine-reciprocals-iii
|
||||
|
||||
# --description--
|
||||
|
||||
In the following equation x, y, and n are positive integers. 1/x + 1/y= 1/n
|
||||
In the following equation $x$, $y$, and $n$ are positive integers.
|
||||
|
||||
<!-- TODO Use MathJax -->
|
||||
$$\frac{1}{x} + \frac{1}{y} = \frac{1}{n}$$
|
||||
|
||||
For a limit L we define F(L) as the number of solutions which satisfy x < y ≤ L.
|
||||
For a limit $L$ we define $F(L)$ as the number of solutions which satisfy $x < y ≤ L$.
|
||||
|
||||
We can verify that F(15) = 4 and F(1000) = 1069. Find F(1012).
|
||||
We can verify that $F(15) = 4$ and $F(1000) = 1069$.
|
||||
|
||||
Find $F({10}^{12})$.
|
||||
|
||||
# --hints--
|
||||
|
||||
`euler454()` should return 5435004633092.
|
||||
`diophantineReciprocalsThree()` should return `5435004633092`.
|
||||
|
||||
```js
|
||||
assert.strictEqual(euler454(), 5435004633092);
|
||||
assert.strictEqual(diophantineReciprocalsThree(), 5435004633092);
|
||||
```
|
||||
|
||||
# --seed--
|
||||
@ -29,12 +31,12 @@ assert.strictEqual(euler454(), 5435004633092);
|
||||
## --seed-contents--
|
||||
|
||||
```js
|
||||
function euler454() {
|
||||
function diophantineReciprocalsThree() {
|
||||
|
||||
return true;
|
||||
}
|
||||
|
||||
euler454();
|
||||
diophantineReciprocalsThree();
|
||||
```
|
||||
|
||||
# --solutions--
|
||||
|
||||
@ -8,18 +8,25 @@ dashedName: problem-455-powers-with-trailing-digits
|
||||
|
||||
# --description--
|
||||
|
||||
Let f(n) be the largest positive integer x less than 109 such that the last 9 digits of nx form the number x (including leading zeros), or zero if no such integer exists.
|
||||
Let $f(n)$ be the largest positive integer $x$ less than ${10}^9$ such that the last 9 digits of $n^x$ form the number $x$ (including leading zeros), or zero if no such integer exists.
|
||||
|
||||
For example:
|
||||
|
||||
f(4) = 411728896 (4411728896 = ...490411728896) f(10) = 0 f(157) = 743757 (157743757 = ...567000743757) Σf(n), 2 ≤ n ≤ 103 = 442530011399 Find Σf(n), 2 ≤ n ≤ 106.
|
||||
$$\begin{align}
|
||||
& f(4) = 411\\,728\\,896 (4^{411\\,728\\,896} = ...490\underline{411728896}) \\\\
|
||||
& f(10) = 0 \\\\
|
||||
& f(157) = 743\\,757 (157^{743\\,757} = ...567\underline{000743757}) \\\\
|
||||
& Σf(n), 2 ≤ n ≤ 103 = 442\\,530\\,011\\,399
|
||||
\end{align}$$
|
||||
|
||||
Find $\sum f(n)$, $2 ≤ n ≤ {10}^6$.
|
||||
|
||||
# --hints--
|
||||
|
||||
`euler455()` should return 450186511399999.
|
||||
`powersWithTrailingDigits()` should return `450186511399999`.
|
||||
|
||||
```js
|
||||
assert.strictEqual(euler455(), 450186511399999);
|
||||
assert.strictEqual(powersWithTrailingDigits(), 450186511399999);
|
||||
```
|
||||
|
||||
# --seed--
|
||||
@ -27,12 +34,12 @@ assert.strictEqual(euler455(), 450186511399999);
|
||||
## --seed-contents--
|
||||
|
||||
```js
|
||||
function euler455() {
|
||||
function powersWithTrailingDigits() {
|
||||
|
||||
return true;
|
||||
}
|
||||
|
||||
euler455();
|
||||
powersWithTrailingDigits();
|
||||
```
|
||||
|
||||
# --solutions--
|
||||
|
||||
@ -8,24 +8,35 @@ dashedName: problem-456-triangles-containing-the-origin-ii
|
||||
|
||||
# --description--
|
||||
|
||||
Define:xn = (1248n mod 32323) - 16161yn = (8421n mod 30103) - 15051
|
||||
Define:
|
||||
|
||||
Pn = {(x1, y1), (x2, y2), ..., (xn, yn)}
|
||||
$$\begin{align}
|
||||
& x_n = ({1248}^n\bmod 32323) - 16161 \\\\
|
||||
& y_n = ({8421}^n\bmod 30103) - 15051 \\\\
|
||||
& P_n = \\{(x_1, y_1), (x_2, y_2), \ldots, (x_n, y_n)\\}
|
||||
\end{align}$$
|
||||
|
||||
For example, P8 = {(-14913, -6630), (-10161, 5625), (5226, 11896), (8340, -10778), (15852, -5203), (-15165, 11295), (-1427, -14495), (12407, 1060)}.
|
||||
For example,
|
||||
$$P_8 = \\{(-14913, -6630), (-10161, 5625), (5226, 11896), (8340, -10778), (15852, -5203), (-15165, 11295), (-1427, -14495), (12407, 1060)\\}$$
|
||||
|
||||
Let C(n) be the number of triangles whose vertices are in Pn which contain the origin in the interior.
|
||||
Let $C(n)$ be the number of triangles whose vertices are in $P_n$ which contain the origin in the interior.
|
||||
|
||||
Examples: C(8) = 20 C(600) = 8950634 C(40 000) = 2666610948988
|
||||
Examples:
|
||||
|
||||
Find C(2 000 000).
|
||||
$$\begin{align}
|
||||
& C(8) = 20 \\\\
|
||||
& C(600) = 8\\,950\\,634 \\\\
|
||||
& C(40\\,000) = 2\\,666\\,610\\,948\\,988
|
||||
\end{align}$$
|
||||
|
||||
Find $C(2\\,000\\,000)$.
|
||||
|
||||
# --hints--
|
||||
|
||||
`euler456()` should return 333333208685971500.
|
||||
`trianglesContainingOriginTwo()` should return `333333208685971500`.
|
||||
|
||||
```js
|
||||
assert.strictEqual(euler456(), 333333208685971500);
|
||||
assert.strictEqual(trianglesContainingOriginTwo(), 333333208685971500);
|
||||
```
|
||||
|
||||
# --seed--
|
||||
@ -33,12 +44,12 @@ assert.strictEqual(euler456(), 333333208685971500);
|
||||
## --seed-contents--
|
||||
|
||||
```js
|
||||
function euler456() {
|
||||
function trianglesContainingOriginTwo() {
|
||||
|
||||
return true;
|
||||
}
|
||||
|
||||
euler456();
|
||||
trianglesContainingOriginTwo();
|
||||
```
|
||||
|
||||
# --solutions--
|
||||
|
||||
@ -8,22 +8,22 @@ dashedName: problem-457-a-polynomial-modulo-the-square-of-a-prime
|
||||
|
||||
# --description--
|
||||
|
||||
Let f(n) = n2 - 3n - 1.
|
||||
Let $f(n) = n^2 - 3n - 1$.
|
||||
|
||||
Let p be a prime.
|
||||
Let $p$ be a prime.
|
||||
|
||||
Let R(p) be the smallest positive integer n such that f(n) mod p2 = 0 if such an integer n exists, otherwise R(p) = 0.
|
||||
Let $R(p)$ be the smallest positive integer $n$ such that $f(n)\bmod p^2 = 0$ if such an integer $n$ exists, otherwise $R(p) = 0$.
|
||||
|
||||
Let SR(L) be ∑R(p) for all primes not exceeding L.
|
||||
Let $SR(L)$ be $\sum R(p)$ for all primes not exceeding $L$.
|
||||
|
||||
Find SR(107).
|
||||
Find $SR({10}^7)$.
|
||||
|
||||
# --hints--
|
||||
|
||||
`euler457()` should return 2647787126797397000.
|
||||
`polynomialModuloSquareOfPrime()` should return `2647787126797397000`.
|
||||
|
||||
```js
|
||||
assert.strictEqual(euler457(), 2647787126797397000);
|
||||
assert.strictEqual(polynomialModuloSquareOfPrime(), 2647787126797397000);
|
||||
```
|
||||
|
||||
# --seed--
|
||||
@ -31,12 +31,12 @@ assert.strictEqual(euler457(), 2647787126797397000);
|
||||
## --seed-contents--
|
||||
|
||||
```js
|
||||
function euler457() {
|
||||
function polynomialModuloSquareOfPrime() {
|
||||
|
||||
return true;
|
||||
}
|
||||
|
||||
euler457();
|
||||
polynomialModuloSquareOfPrime();
|
||||
```
|
||||
|
||||
# --solutions--
|
||||
|
||||
@ -8,20 +8,20 @@ dashedName: problem-458-permutations-of-project
|
||||
|
||||
# --description--
|
||||
|
||||
Consider the alphabet A made out of the letters of the word "project": A={c,e,j,o,p,r,t}.
|
||||
Consider the alphabet $A$ made out of the letters of the word `project`: $A = \\{c, e, j, o, p, r, t\\}$.
|
||||
|
||||
Let T(n) be the number of strings of length n consisting of letters from A that do not have a substring that is one of the 5040 permutations of "project".
|
||||
Let $T(n)$ be the number of strings of length $n$ consisting of letters from $A$ that do not have a substring that is one of the 5040 permutations of `project`.
|
||||
|
||||
T(7)=77-7!=818503.
|
||||
$T(7) = 7^7 - 7! = 818\\,503$.
|
||||
|
||||
Find T(1012). Give the last 9 digits of your answer.
|
||||
Find $T({10}^{12})$. Give the last 9 digits of your answer.
|
||||
|
||||
# --hints--
|
||||
|
||||
`euler458()` should return 423341841.
|
||||
`permutationsOfProject()` should return `423341841`.
|
||||
|
||||
```js
|
||||
assert.strictEqual(euler458(), 423341841);
|
||||
assert.strictEqual(permutationsOfProject(), 423341841);
|
||||
```
|
||||
|
||||
# --seed--
|
||||
@ -29,12 +29,12 @@ assert.strictEqual(euler458(), 423341841);
|
||||
## --seed-contents--
|
||||
|
||||
```js
|
||||
function euler458() {
|
||||
function permutationsOfProject() {
|
||||
|
||||
return true;
|
||||
}
|
||||
|
||||
euler458();
|
||||
permutationsOfProject();
|
||||
```
|
||||
|
||||
# --solutions--
|
||||
|
||||
@ -8,28 +8,38 @@ dashedName: problem-459-flipping-game
|
||||
|
||||
# --description--
|
||||
|
||||
The flipping game is a two player game played on a N by N square board.
|
||||
The flipping game is a two player game played on a $N$ by $N$ square board.
|
||||
|
||||
Each square contains a disk with one side white and one side black.
|
||||
|
||||
The game starts with all disks showing their white side.
|
||||
|
||||
A turn consists of flipping all disks in a rectangle with the following properties: the upper right corner of the rectangle contains a white disk the rectangle width is a perfect square (1, 4, 9, 16, ...) the rectangle height is a triangular number (1, 3, 6, 10, ...)
|
||||
A turn consists of flipping all disks in a rectangle with the following properties:
|
||||
|
||||
- the upper right corner of the rectangle contains a white disk
|
||||
- the rectangle width is a perfect square (1, 4, 9, 16, ...)
|
||||
- the rectangle height is a triangular number (1, 3, 6, 10, ...)
|
||||
|
||||
<img class="img-responsive center-block" alt="flipping all disks in a 4x3 rectangle on a 5x5 board" src="https://cdn.freecodecamp.org/curriculum/project-euler/flipping-game-1.png" style="background-color: white; padding: 10px;">
|
||||
|
||||
Players alternate turns. A player wins by turning the grid all black.
|
||||
|
||||
Let W(N) be the number of winning moves for the first player on a N by N board with all disks white, assuming perfect play. W(1) = 1, W(2) = 0, W(5) = 8 and W(102) = 31395.
|
||||
Let $W(N)$ be the number of winning moves for the first player on a $N$ by $N$ board with all disks white, assuming perfect play.
|
||||
|
||||
For N=5, the first player's eight winning first moves are:
|
||||
$W(1) = 1$, $W(2) = 0$, $W(5) = 8$ and $W({10}^2) = 31\\,395$.
|
||||
|
||||
Find W(106).
|
||||
For $N = 5$, the first player's eight winning first moves are:
|
||||
|
||||
<img class="img-responsive center-block" alt="eight winning first moves for N = 5" src="https://cdn.freecodecamp.org/curriculum/project-euler/flipping-game-2.png" style="background-color: white; padding: 10px;">
|
||||
|
||||
Find $W({10}^6)$.
|
||||
|
||||
# --hints--
|
||||
|
||||
`euler459()` should return 3996390106631.
|
||||
`flippingGame()` should return `3996390106631`.
|
||||
|
||||
```js
|
||||
assert.strictEqual(euler459(), 3996390106631);
|
||||
assert.strictEqual(flippingGame(), 3996390106631);
|
||||
```
|
||||
|
||||
# --seed--
|
||||
@ -37,12 +47,12 @@ assert.strictEqual(euler459(), 3996390106631);
|
||||
## --seed-contents--
|
||||
|
||||
```js
|
||||
function euler459() {
|
||||
function flippingGame() {
|
||||
|
||||
return true;
|
||||
}
|
||||
|
||||
euler459();
|
||||
flippingGame();
|
||||
```
|
||||
|
||||
# --solutions--
|
||||
|
||||
@ -8,24 +8,33 @@ dashedName: problem-460-an-ant-on-the-move
|
||||
|
||||
# --description--
|
||||
|
||||
On the Euclidean plane, an ant travels from point A(0, 1) to point B(d, 1) for an integer d.
|
||||
On the Euclidean plane, an ant travels from point $A(0, 1)$ to point $B(d, 1)$ for an integer $d$.
|
||||
|
||||
In each step, the ant at point (x0, y0) chooses one of the lattice points (x1, y1) which satisfy x1 ≥ 0 and y1 ≥ 1 and goes straight to (x1, y1) at a constant velocity v. The value of v depends on y0 and y1 as follows: If y0 = y1, the value of v equals y0. If y0 ≠ y1, the value of v equals (y1 - y0) / (ln(y1) - ln(y0)).
|
||||
In each step, the ant at point ($x_0$, $y_0$) chooses one of the lattice points ($x_1$, $y_1$) which satisfy $x_1 ≥ 0$ and $y_1 ≥ 1$ and goes straight to ($x_1$, $y_1$) at a constant velocity $v$. The value of $v$ depends on $y_0$ and $y_1$ as follows:
|
||||
|
||||
The left image is one of the possible paths for d = 4. First the ant goes from A(0, 1) to P1(1, 3) at velocity (3 - 1) / (ln(3) - ln(1)) ≈ 1.8205. Then the required time is sqrt(5) / 1.8205 ≈ 1.2283. From P1(1, 3) to P2(3, 3) the ant travels at velocity 3 so the required time is 2 / 3 ≈ 0.6667. From P2(3, 3) to B(4, 1) the ant travels at velocity (1 - 3) / (ln(1) - ln(3)) ≈ 1.8205 so the required time is sqrt(5) / 1.8205 ≈ 1.2283. Thus the total required time is 1.2283 + 0.6667 + 1.2283 = 3.1233.
|
||||
- If $y_0 = y_1$, the value of $v$ equals $y_0$.
|
||||
- If $y_0 ≠ y_1$, the value of $v$ equals $\frac{y_1 - y_0}{\ln y_1 - \ln y_0}$.
|
||||
|
||||
The right image is another path. The total required time is calculated as 0.98026 + 1 + 0.98026 = 2.96052. It can be shown that this is the quickest path for d = 4.
|
||||
The left image is one of the possible paths for $d = 4$. First the ant goes from $A(0, 1)$ to $P_1(1, 3)$ at velocity $\frac{3 - 1}{\ln 3 - \ln 1} ≈ 1.8205$. Then the required time is $\frac{\sqrt{5}}{1.820} ≈ 1.2283$.
|
||||
|
||||
Let F(d) be the total required time if the ant chooses the quickest path. For example, F(4) ≈ 2.960516287. We can verify that F(10) ≈ 4.668187834 and F(100) ≈ 9.217221972.
|
||||
From $P_1(1, 3)$ to $P_2(3, 3)$ the ant travels at velocity 3 so the required time is $\frac{2}{3} ≈ 0.6667$. From $P_2(3, 3)$ to $B(4, 1)$ the ant travels at velocity $\frac{1 - 3}{\ln 1 - \ln 3} ≈ 1.8205$ so the required time is $\frac{\sqrt{5}}{1.8205} ≈ 1.2283$.
|
||||
|
||||
Find F(10000). Give your answer rounded to nine decimal places.
|
||||
Thus the total required time is $1.2283 + 0.6667 + 1.2283 = 3.1233$.
|
||||
|
||||
The right image is another path. The total required time is calculated as $0.98026 + 1 + 0.98026 = 2.96052$. It can be shown that this is the quickest path for $d = 4$.
|
||||
|
||||
<img class="img-responsive center-block" alt="two possible paths for d = 4" src="https://cdn.freecodecamp.org/curriculum/project-euler/an-ant-on-the-move.jpg" style="background-color: white; padding: 10px;">
|
||||
|
||||
Let $F(d)$ be the total required time if the ant chooses the quickest path. For example, $F(4) ≈ 2.960\\,516\\,287$. We can verify that $F(10) ≈ 4.668\\,187\\,834$ and $F(100) ≈ 9.217\\,221\\,972$.
|
||||
|
||||
Find $F(10\\,000)$. Give your answer rounded to nine decimal places.
|
||||
|
||||
# --hints--
|
||||
|
||||
`euler460()` should return 18.420738199.
|
||||
`antOnTheMove()` should return `18.420738199`.
|
||||
|
||||
```js
|
||||
assert.strictEqual(euler460(), 18.420738199);
|
||||
assert.strictEqual(antOnTheMove(), 18.420738199);
|
||||
```
|
||||
|
||||
# --seed--
|
||||
@ -33,12 +42,12 @@ assert.strictEqual(euler460(), 18.420738199);
|
||||
## --seed-contents--
|
||||
|
||||
```js
|
||||
function euler460() {
|
||||
function antOnTheMove() {
|
||||
|
||||
return true;
|
||||
}
|
||||
|
||||
euler460();
|
||||
antOnTheMove();
|
||||
```
|
||||
|
||||
# --solutions--
|
||||
|
||||
Reference in New Issue
Block a user