261 lines
6.3 KiB
Plaintext
261 lines
6.3 KiB
Plaintext
{
|
||
"cells": [
|
||
{
|
||
"cell_type": "markdown",
|
||
"metadata": {},
|
||
"source": [
|
||
"# Euler Problem 31\n",
|
||
"\n",
|
||
"In England the currency is made up of pound, £, and pence, p, and there are eight coins in general circulation:\n",
|
||
"\n",
|
||
"1p, 2p, 5p, 10p, 20p, 50p, £1 (100p) and £2 (200p).\n",
|
||
"\n",
|
||
"It is possible to make £2 in the following way:\n",
|
||
"\n",
|
||
"1×£1 + 1×50p + 2×20p + 1×5p + 1×2p + 3×1p\n",
|
||
"\n",
|
||
"How many different ways can £2 be made using any number of coins?"
|
||
]
|
||
},
|
||
{
|
||
"cell_type": "markdown",
|
||
"metadata": {
|
||
"collapsed": true
|
||
},
|
||
"source": [
|
||
"Let's do it non-recursive. Tricky part is to use ceil, otherwise we get wrong ranges for example for $\\frac{50}{20}$."
|
||
]
|
||
},
|
||
{
|
||
"cell_type": "code",
|
||
"execution_count": 1,
|
||
"metadata": {},
|
||
"outputs": [
|
||
{
|
||
"name": "stdout",
|
||
"output_type": "stream",
|
||
"text": [
|
||
"[0, 1]\n",
|
||
"[0, 1, 2]\n"
|
||
]
|
||
}
|
||
],
|
||
"source": [
|
||
"from math import ceil\n",
|
||
"\n",
|
||
"print(list(range(50 // 20)))\n",
|
||
"print(list(range(ceil(50 / 20))))"
|
||
]
|
||
},
|
||
{
|
||
"cell_type": "code",
|
||
"execution_count": 8,
|
||
"metadata": {},
|
||
"outputs": [],
|
||
"source": [
|
||
"from math import ceil\n",
|
||
"\n",
|
||
"def calculate():\n",
|
||
" r = 200\n",
|
||
" c = 0\n",
|
||
"\n",
|
||
" if r % 200 == 0:\n",
|
||
" c += 1\n",
|
||
" for p200 in range(ceil(r / 200)):\n",
|
||
" r200 = r - p200 * 200\n",
|
||
" if r200 % 100 == 0:\n",
|
||
" c += 1\n",
|
||
" for p100 in range(ceil(r200 / 100)):\n",
|
||
" r100 = r200 - p100 * 100\n",
|
||
" if r100 % 50 == 0:\n",
|
||
" c += 1\n",
|
||
" for p50 in range(ceil(r100 / 50)):\n",
|
||
" r50 = r100 - p50 * 50\n",
|
||
" if r50 % 20 == 0:\n",
|
||
" c += 1\n",
|
||
" for p20 in range(ceil(r50 / 20)):\n",
|
||
" r20 = r50 - p20 * 20\n",
|
||
" if r20 <= 0:\n",
|
||
" break\n",
|
||
" if r20 % 10 == 0:\n",
|
||
" c += 1\n",
|
||
" for p10 in range(ceil(r20 / 10)):\n",
|
||
" r10 = r20 - p10 * 10\n",
|
||
" if r10 % 5 == 0:\n",
|
||
" c += 1\n",
|
||
" for p5 in range(ceil(r10 / 5)):\n",
|
||
" r5 = r10 - p5 * 5\n",
|
||
" if r5 % 2 == 0:\n",
|
||
" c += 1\n",
|
||
" for p2 in range(ceil(r5 / 2)):\n",
|
||
" r2 = r5 - p2 * 2\n",
|
||
" if r2 % 1 == 0:\n",
|
||
" c += 1\n",
|
||
"\n",
|
||
" return c\n",
|
||
"\n",
|
||
"s = calculate()"
|
||
]
|
||
},
|
||
{
|
||
"cell_type": "code",
|
||
"execution_count": 9,
|
||
"metadata": {},
|
||
"outputs": [
|
||
{
|
||
"name": "stdout",
|
||
"output_type": "stream",
|
||
"text": [
|
||
"73682\n"
|
||
]
|
||
}
|
||
],
|
||
"source": [
|
||
"print(s)\n",
|
||
"assert(s == 73682)"
|
||
]
|
||
},
|
||
{
|
||
"cell_type": "markdown",
|
||
"metadata": {},
|
||
"source": [
|
||
"I came up with a more concise solution that can be implemented in Scheme. I would like to compare the performance of the solutions."
|
||
]
|
||
},
|
||
{
|
||
"cell_type": "code",
|
||
"execution_count": 34,
|
||
"metadata": {},
|
||
"outputs": [
|
||
{
|
||
"name": "stdout",
|
||
"output_type": "stream",
|
||
"text": [
|
||
"73682\n"
|
||
]
|
||
}
|
||
],
|
||
"source": [
|
||
"def count_change(change, coins):\n",
|
||
" if not coins:\n",
|
||
" return 0\n",
|
||
" count = 0\n",
|
||
" coin, coins = coins[0], coins[1:]\n",
|
||
" for i in range(change // coin + 1):\n",
|
||
" if change - i * coin == 0:\n",
|
||
" count += 1\n",
|
||
" continue\n",
|
||
" if change - i * coin < 0:\n",
|
||
" break\n",
|
||
" count += count_change(change - i * coin, coins)\n",
|
||
" return count\n",
|
||
"\n",
|
||
"def calculate_iterative():\n",
|
||
" return count_change(200, [200, 100, 50, 20, 10, 5, 2, 1])\n",
|
||
"\n",
|
||
"s = calculate_iterative()\n",
|
||
"print(s)"
|
||
]
|
||
},
|
||
{
|
||
"cell_type": "code",
|
||
"execution_count": 35,
|
||
"metadata": {},
|
||
"outputs": [
|
||
{
|
||
"name": "stdout",
|
||
"output_type": "stream",
|
||
"text": [
|
||
"0.16300111500095227\n",
|
||
"11.429953110000497\n"
|
||
]
|
||
}
|
||
],
|
||
"source": [
|
||
"import timeit\n",
|
||
"print(timeit.timeit(calculate, number=10))\n",
|
||
"print(timeit.timeit(calculate_iterative, number=10))"
|
||
]
|
||
},
|
||
{
|
||
"cell_type": "markdown",
|
||
"metadata": {},
|
||
"source": [
|
||
"Turns out I was way smarter than I thought when I have developped calculate. It checks the remainder and by doing that saves a ton of iterations."
|
||
]
|
||
},
|
||
{
|
||
"cell_type": "code",
|
||
"execution_count": 36,
|
||
"metadata": {},
|
||
"outputs": [
|
||
{
|
||
"name": "stdout",
|
||
"output_type": "stream",
|
||
"text": [
|
||
"73682\n",
|
||
"0.34602016500139143\n"
|
||
]
|
||
}
|
||
],
|
||
"source": [
|
||
"def count_change(change, coins):\n",
|
||
" count = 0\n",
|
||
" coin, coins = coins[0], coins[1:]\n",
|
||
"\n",
|
||
" if change % coin == 0:\n",
|
||
" count += 1\n",
|
||
" \n",
|
||
" if not coins:\n",
|
||
" return count\n",
|
||
" \n",
|
||
" for i in range(ceil(change / coin)):\n",
|
||
" count += count_change(change - i * coin, coins)\n",
|
||
" return count\n",
|
||
"\n",
|
||
"def calculate_iterative():\n",
|
||
" return count_change(200, [200, 100, 50, 20, 10, 5, 2, 1])\n",
|
||
"\n",
|
||
"s = calculate_iterative()\n",
|
||
"print(s)\n",
|
||
"print(timeit.timeit(calculate_iterative, number=10))"
|
||
]
|
||
},
|
||
{
|
||
"cell_type": "code",
|
||
"execution_count": null,
|
||
"metadata": {
|
||
"collapsed": true
|
||
},
|
||
"outputs": [],
|
||
"source": []
|
||
}
|
||
],
|
||
"metadata": {
|
||
"completion_date": "Fri, 25 Aug 2017, 13:02",
|
||
"kernelspec": {
|
||
"display_name": "Python 3",
|
||
"language": "python3.6",
|
||
"name": "python3"
|
||
},
|
||
"language_info": {
|
||
"codemirror_mode": {
|
||
"name": "ipython",
|
||
"version": 3
|
||
},
|
||
"file_extension": ".py",
|
||
"mimetype": "text/x-python",
|
||
"name": "python",
|
||
"nbconvert_exporter": "python",
|
||
"pygments_lexer": "ipython3",
|
||
"version": "3.6.5"
|
||
},
|
||
"tags": [
|
||
"recursion",
|
||
"coins"
|
||
]
|
||
},
|
||
"nbformat": 4,
|
||
"nbformat_minor": 2
|
||
}
|