139 lines
3.2 KiB
Plaintext
139 lines
3.2 KiB
Plaintext
{
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"cells": [
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{
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"cell_type": "markdown",
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"id": "eb246b50",
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"metadata": {},
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"source": [
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"# [Digit Cancelling Fractions](https://projecteuler.net/problem=33)\n",
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"\n",
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"The search term for this concept is an [anomalous cancellation](https://en.wikipedia.org/wiki/Anomalous_cancellation). We can write a function that checks if an anomalous cancellation can happen by storing the digits of the numerator and denominator into four variables total, then checking four separate cases for if a digit can be \"cancelled.\""
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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": "d4e94963",
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"metadata": {},
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"outputs": [],
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"source": [
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"def digits(n):\n",
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" return (n // 10, n % 10)\n",
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"\n",
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"\n",
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"def can_cancel_digits(n, d):\n",
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" x, y = digits(n)\n",
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" z, w = digits(d)\n",
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"\n",
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" f = QQ(n/d)\n",
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"\n",
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" if x == z and w != 0 and y/w == f:\n",
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" return True\n",
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" elif x == w and y/z == f:\n",
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" return True\n",
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" elif y == z and w != 0 and x/w == f:\n",
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" return True\n",
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" elif y == w and w != 0 and x/z == f:\n",
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" return True\n",
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"\n",
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" return False"
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]
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},
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{
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"cell_type": "markdown",
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"id": "a68309e9",
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"metadata": {},
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"source": [
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"We're only dealing with two-digit numerators and denominators, so this is easy to brute force."
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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": "756614f7",
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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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"{(16, 64), (19, 95), (26, 65), (49, 98)}"
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]
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},
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"execution_count": 2,
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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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"fractions = set()\n",
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"for n in range(10, 100):\n",
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" for d in range(n + 1, 100):\n",
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" if can_cancel_digits(n, d):\n",
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" fractions.add((n, d))\n",
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"\n",
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"fractions"
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]
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},
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{
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"cell_type": "markdown",
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"id": "6d3fb43b",
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"metadata": {},
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"source": [
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"There's only four such fractions."
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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": "90f47a7f",
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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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"100"
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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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"prod(QQ(n/d) for (n, d) in fractions).denominator()"
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]
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},
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{
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"cell_type": "markdown",
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"id": "1514f872",
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"metadata": {},
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"source": [
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"#### Copyright (C) 2025 filifa\n",
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"\n",
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"This work is licensed under the [Creative Commons Attribution-ShareAlike 4.0 International license](https://creativecommons.org/licenses/by-sa/4.0/) and the [BSD Zero Clause license](https://spdx.org/licenses/0BSD.html)."
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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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