add problem 100
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filifa
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"cells": [
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{
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"cell_type": "markdown",
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"id": "4945116b",
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"metadata": {},
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"source": [
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"# [Arranged Probability](https://projecteuler.net/problem=100)\n",
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"\n",
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"We can rephrase this problem as finding integer solutions to\n",
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"$$\\frac{b}{t} \\times \\frac{b-1}{t-1} = \\frac{1}{2}$$\n",
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"Specifically, we're looking for the first solution with $t > 10^{12}$. This is a [Diophantine problem](https://en.wikipedia.org/wiki/Diophantine_equation). "
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]
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},
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{
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"cell_type": "code",
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"execution_count": 1,
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"id": "fa08d031",
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"metadata": {},
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"outputs": [
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{
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"data": {
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"text/plain": [
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"[(-1/8*(12*sqrt(2) + 17)^t*(sqrt(2) + 2) + 1/8*(-12*sqrt(2) + 17)^t*(sqrt(2) - 2) + 1/2,\n",
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" -1/8*sqrt(2)*((12*sqrt(2) + 17)^t*(sqrt(2) + 2) + (-12*sqrt(2) + 17)^t*(sqrt(2) - 2)) + 1/2),\n",
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" (-1/8*(12*sqrt(2) + 17)^t*(7*sqrt(2) + 10) + 1/8*(-12*sqrt(2) + 17)^t*(7*sqrt(2) - 10) + 1/2,\n",
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" -1/8*sqrt(2)*((12*sqrt(2) + 17)^t*(7*sqrt(2) + 10) + (-12*sqrt(2) + 17)^t*(7*sqrt(2) - 10)) + 1/2),\n",
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" (1/8*(12*sqrt(2) + 17)^t*(sqrt(2) + 2) - 1/8*(-12*sqrt(2) + 17)^t*(sqrt(2) - 2) + 1/2,\n",
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" 1/8*sqrt(2)*((12*sqrt(2) + 17)^t*(sqrt(2) + 2) + (-12*sqrt(2) + 17)^t*(sqrt(2) - 2)) + 1/2),\n",
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" (1/8*(12*sqrt(2) + 17)^t*(7*sqrt(2) + 10) - 1/8*(-12*sqrt(2) + 17)^t*(7*sqrt(2) - 10) + 1/2,\n",
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" 1/8*sqrt(2)*((12*sqrt(2) + 17)^t*(7*sqrt(2) + 10) + (-12*sqrt(2) + 17)^t*(7*sqrt(2) - 10)) + 1/2)]"
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]
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},
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"execution_count": 1,
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"metadata": {},
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"output_type": "execute_result"
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}
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],
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"source": [
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"var('b,t')\n",
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"\n",
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"sols = sorted(solve_diophantine(b/t * (b-1)/(t-1) == 1/2))\n",
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"sols"
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]
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},
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{
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"cell_type": "markdown",
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"id": "59126a72",
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"metadata": {},
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"source": [
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"Inspecting these parameterizations, the first two will always generate negative solutions for integer $t$, so we'll only consider the last two. We'll evaluate them at $t=0,1,2,\\ldots$ to find the first instance where the total number of discs is greater than $10^{12}$."
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]
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},
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{
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"cell_type": "code",
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"execution_count": 2,
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"id": "4f211b1a",
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"metadata": {},
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"outputs": [],
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"source": [
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"from itertools import count\n",
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"\n",
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"candidates = []\n",
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"for sol in sols[2:4]:\n",
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" for n in count(0):\n",
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" b, t = sol[0](t=n), sol[1](t=n)\n",
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" if t > 10^12:\n",
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" candidates.append((b, t))\n",
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" break"
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]
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},
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{
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"cell_type": "markdown",
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"id": "ba65bcf6",
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"metadata": {},
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"source": [
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"Then we'll select the smaller $t$ of the two."
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]
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},
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{
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"cell_type": "code",
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"execution_count": 3,
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"id": "9c06a2c4",
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"metadata": {},
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"outputs": [
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{
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"data": {
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"text/plain": [
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"[(756872327473, 1070379110497), (4411375203411, 6238626641380)]"
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]
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},
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"execution_count": 3,
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"metadata": {},
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"output_type": "execute_result"
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}
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],
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"source": [
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"[(int(b), int(t)) for (b, t) in candidates]"
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]
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},
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{
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"cell_type": "markdown",
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"id": "15f57948",
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"metadata": {},
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"source": [
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"This tells us that $b=756872327473, t=1070379110497$ is the arrangement we're after.\n",
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"\n",
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"## Alternate method\n",
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"However, we don't need SageMath's `solve_diophantine` for this problem. In fact, we can solve it using the same techniques as [problem 45](https://projecteuler.net/problem=45).\n",
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"\n",
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"First, rearrange:\n",
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"$$2b^2 - 2b - t^2 + t = 0$$\n",
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"Then complete the square:\n",
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"$$(2t-1)^2 - 2(2b-1)^2 = -1$$\n",
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"Then substitute $X=2t-1$ and $Y=2b-1$ to get a Pell equation:\n",
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"$$X^2 - 2Y^2 = -1$$\n",
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"The problem gives $b=85, t=120$ as a solution to the original equation; this corresponds to $X=239, Y=169$ for our Pell equation. To generate more solutions, we'll need a fundamental solution to this Pell equation, which we can find with SageMath, or with continued fractions, as in [problem 66](https://projecteuler.net/problem=66)."
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]
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},
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{
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"cell_type": "code",
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"execution_count": 4,
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"id": "0349d6a5",
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"metadata": {},
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"outputs": [
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{
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"data": {
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"text/plain": [
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"sqrt2 + 1"
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]
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},
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"execution_count": 4,
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"metadata": {},
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"output_type": "execute_result"
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}
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],
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"source": [
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"K.<sqrt2> = QQ[sqrt(2)]\n",
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"u, *_ = K.units()\n",
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"u"
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]
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},
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{
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"cell_type": "markdown",
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"id": "2e30da3f",
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"metadata": {},
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"source": [
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"With both of these values in hand, we can easily generate more solutions to $X^2 - 2Y^2 = -1$ by computing $\\left(239+169\\sqrt{2}\\right)\\left(1+\\sqrt{2}\\right)^k$ for successive values of $k$. Then we'll check to see if they correspond to solutions to our original equation."
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]
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},
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{
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"cell_type": "code",
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"execution_count": 5,
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"id": "989c878e",
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"metadata": {},
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"outputs": [],
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"source": [
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"z = 239 + 169*sqrt2\n",
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"while True:\n",
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" z *= u\n",
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" if z.norm() != -1:\n",
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" continue\n",
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" \n",
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" x, y = z.parts()\n",
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" if (x + 1) % 2 != 0 or (y + 1) % 2 != 0:\n",
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" continue\n",
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" \n",
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" t, b = (x + 1) // 2, (y + 1) // 2\n",
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" if t > 10^12:\n",
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" break"
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]
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},
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{
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"cell_type": "markdown",
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"id": "02b30f6e",
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"metadata": {},
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"source": [
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"As expected, we get the same solution as above."
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]
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},
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{
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"cell_type": "code",
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"execution_count": 6,
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"id": "ac482fb0",
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"metadata": {},
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"outputs": [
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{
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"data": {
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"text/plain": [
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"(756872327473, 1070379110497)"
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]
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},
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"execution_count": 6,
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"metadata": {},
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"output_type": "execute_result"
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}
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],
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"source": [
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"b, t"
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]
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}
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],
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"metadata": {
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"kernelspec": {
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"display_name": "SageMath 9.5",
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"language": "sage",
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"name": "sagemath"
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},
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"language_info": {
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"codemirror_mode": {
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"name": "ipython",
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"version": 3
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},
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"file_extension": ".py",
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"mimetype": "text/x-python",
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"name": "python",
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"nbconvert_exporter": "python",
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"pygments_lexer": "ipython3",
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"version": "3.11.2"
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}
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},
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"nbformat": 4,
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"nbformat_minor": 5
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}
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