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@ -4,7 +4,7 @@
"cell_type": "markdown", "cell_type": "markdown",
"metadata": {}, "metadata": {},
"source": [ "source": [
"<div align=\"right\">Peter Norvig 2014</i></div>\n", "<div align=\"right\">Peter Norvig 2014</div>\n",
"\n", "\n",
"# Cryptarithmetic (Alphametic) Problems\n", "# Cryptarithmetic (Alphametic) Problems\n",
"\n", "\n",
@ -25,10 +25,11 @@
"6. Should I find one solution (faster) or all solutions (more complete)? I'll handle both use cases by \n", "6. Should I find one solution (faster) or all solutions (more complete)? I'll handle both use cases by \n",
"implementing my function `solve` to return an iterator, which yields solutions one at a time; you can get the first one with `next` or all of them with `set`. \n", "implementing my function `solve` to return an iterator, which yields solutions one at a time; you can get the first one with `next` or all of them with `set`. \n",
"\n", "\n",
"## The solution: `solve`\n", "# The solution: `solve`\n",
"\n", "\n",
"Below we see that `solve` works by generating every way to replace letters in the formula with numbers,\n", "Below we see that `solve` works by generating every way to replace letters in the formula with numbers,\n",
"and then filtering them to keep only valid strings (ones that evaluate to true and have no leading zero)." "and then filtering them to keep only valid strings (ones that evaluate to true and have no leading zero).\n",
"The `str.translate` method is used to do the replacements."
] ]
}, },
{ {
@ -43,22 +44,25 @@
"def solve(formula):\n", "def solve(formula):\n",
" \"\"\"Given a formula like 'NUM + BER = PLAY', fill in digits to solve it.\n", " \"\"\"Given a formula like 'NUM + BER = PLAY', fill in digits to solve it.\n",
" Generate all valid digit-filled-in strings.\"\"\"\n", " Generate all valid digit-filled-in strings.\"\"\"\n",
" return filter(valid, replace_letters(formula.replace(' = ', ' == ')))\n", " return filter(valid, letter_replacements(formula))\n",
"\n", "\n",
"def replace_letters(formula):\n", "def letter_replacements(formula):\n",
" \"\"\"Generate all possible replacements of letters with digits in formula.\"\"\"\n", " \"\"\"All possible replacements of letters with digits in formula.\"\"\"\n",
" letters = ''.join(set(re.findall('[A-Z]', formula)))\n", " formula = formula.replace(' = ', ' == ') # Allow = or ==\n",
" letters = cat(set(re.findall('[A-Z]', formula)))\n",
" for digits in itertools.permutations('1234567890', len(letters)):\n", " for digits in itertools.permutations('1234567890', len(letters)):\n",
" yield formula.translate(str.maketrans(letters, ''.join(digits)))\n", " yield formula.translate(str.maketrans(letters, cat(digits)))\n",
"\n", "\n",
"def valid(exp):\n", "def valid(exp):\n",
" \"\"\"Expression is valid iff it has no leading zero, and evaluates to true.\"\"\"\n", " \"\"\"Expression is valid iff it has no leading zero, and evaluates to true.\"\"\"\n",
" try:\n", " try:\n",
" return not leading_zero(exp) and eval(exp)\n", " return not leading_zero(exp) and eval(exp) is True\n",
" except ArithmeticError:\n", " except ArithmeticError:\n",
" return False\n", " return False\n",
" \n", " \n",
"leading_zero = re.compile(r'\\b0[0-9]').search" "cat = ''.join # Function to concatenate strings\n",
" \n",
"leading_zero = re.compile(r'\\b0[0-9]').search # Function to check for illegal number"
] ]
}, },
{ {
@ -69,7 +73,7 @@
{ {
"data": { "data": {
"text/plain": [ "text/plain": [
"'746 + 289 == 1035'" "'489 + 537 == 1026'"
] ]
}, },
"execution_count": 2, "execution_count": 2,
@ -90,8 +94,8 @@
"name": "stdout", "name": "stdout",
"output_type": "stream", "output_type": "stream",
"text": [ "text": [
"CPU times: user 17 s, sys: 68.2 ms, total: 17.1 s\n", "CPU times: user 17.2 s, sys: 61.3 ms, total: 17.3 s\n",
"Wall time: 17.2 s\n" "Wall time: 17.4 s\n"
] ]
}, },
{ {
@ -113,29 +117,52 @@
"cell_type": "markdown", "cell_type": "markdown",
"metadata": {}, "metadata": {},
"source": [ "source": [
"So there are 96 solutions, but `solve` is a bit slow to find them all. How could we make `solve` faster?\n", "# A faster solution: `faster_solve`\n",
"\n", "\n",
"## A faster solution: `faster_solve`\n", "Depending on your computer, that probably took 15 or 20 seconds. Can we make it faster? To answer the question, I start by profiling to see where the time is spent. I can use the magic function `%prun` to profile:"
"\n", ]
"I used `%prun` to get profiling results, and saw that 2/3 of the time is spent in `eval`. So let's eliminate the calls to `eval`. That should be doable, because the expression we are evaluating is basically the same each time, but with different permutations of digits filled in. We could save a lot of work if we convert the expression into a Python function, compile that function just once, and then call the function for each permutation of digits. In other words, we want to take an expression such as\n", },
{
"cell_type": "code",
"execution_count": 4,
"metadata": {},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
" "
]
}
],
"source": [
"%prun next(solve('NUM + BER = PLAY'))"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"We see that about 2/3 of the time is spent in `eval`. So let's eliminate the calls to `eval`. That should be doable, because the expression we are evaluating is basically the same each time, but with different permutations of digits filled in. We could save a lot of work if we convert the expression into a Python function, compile that function just once, and then call the function for each of the 3.6 million permutations of digits. We want to take an expression such as:\n",
"\n", "\n",
" \"NUM + BER == PLAY\"\n", " \"NUM + BER == PLAY\"\n",
" \n", " \n",
"and transform it into the Python function\n", "and transform it into the Python function:\n",
"\n", "\n",
"\n", " (lambda A,B,E,L,M,N,P,R,U,Y: \n",
" lambda A,B,E,L,M,N,P,R,U,Y: (100*N+10*U+M) + (100*B+10*E+R) == (1000*P+100*L+10*A+Y)\n", " (100*N+10*U+M) + (100*B+10*E+R) == (1000*P+100*L+10*A+Y))\n",
" \n", " \n",
"Actually that's not quite right. The rules say that \"N\", \"B\", and \"P\" cannot be zero. So the function should be:\n", "Actually that's not quite right. The rules say that \"N\", \"B\", and \"P\" cannot be zero. So the function should be:\n",
"\n", "\n",
" A,B,E,L,M,N,P,R,U,Y: B and N and P and ((100*N+10*U+M) + (100*B+10*E+R) == (1000*P+100*L+10*A+Y))\n", " (lambda A,B,E,L,M,N,P,R,U,Y: \n",
" B and N and P and ((100*N+10*U+M) + (100*B+10*E+R) == (1000*P+100*L+10*A+Y)))\n",
"\n", "\n",
"Here is the code to compile a formula into a Python function: " "Here is the code to compile a formula into a Python function: "
] ]
}, },
{ {
"cell_type": "code", "cell_type": "code",
"execution_count": 4, "execution_count": 5,
"metadata": {}, "metadata": {},
"outputs": [], "outputs": [],
"source": [ "source": [
@ -144,7 +171,7 @@
" in same order as parms of function. For example, 'YOU == ME**2' returns\n", " in same order as parms of function. For example, 'YOU == ME**2' returns\n",
" (lambda E,M,O,U,Y: M and Y and ((100*Y+10*O+U) == (10*M+E)**2), 'YMEUO'\"\"\"\n", " (lambda E,M,O,U,Y: M and Y and ((100*Y+10*O+U) == (10*M+E)**2), 'YMEUO'\"\"\"\n",
" formula = formula.replace(' = ', ' == ')\n", " formula = formula.replace(' = ', ' == ')\n",
" letters = ''.join(sorted(set(re.findall('[A-Z]', formula))))\n", " letters = cat(sorted(set(re.findall('[A-Z]', formula))))\n",
" firstletters = sorted(set(re.findall(r'\\b([A-Z])[A-Z]', formula)))\n", " firstletters = sorted(set(re.findall(r'\\b([A-Z])[A-Z]', formula)))\n",
" body = re.sub('[A-Z]+', compile_word, formula)\n", " body = re.sub('[A-Z]+', compile_word, formula)\n",
" body = ' and '.join(firstletters + [body])\n", " body = ' and '.join(firstletters + [body])\n",
@ -164,7 +191,7 @@
}, },
{ {
"cell_type": "code", "cell_type": "code",
"execution_count": 5, "execution_count": 6,
"metadata": {}, "metadata": {},
"outputs": [ "outputs": [
{ {
@ -180,7 +207,7 @@
"(<function __main__.<lambda>(E, M, O, U, Y)>, 'EMOUY')" "(<function __main__.<lambda>(E, M, O, U, Y)>, 'EMOUY')"
] ]
}, },
"execution_count": 5, "execution_count": 6,
"metadata": {}, "metadata": {},
"output_type": "execute_result" "output_type": "execute_result"
} }
@ -191,7 +218,7 @@
}, },
{ {
"cell_type": "code", "cell_type": "code",
"execution_count": 6, "execution_count": 7,
"metadata": {}, "metadata": {},
"outputs": [ "outputs": [
{ {
@ -207,7 +234,7 @@
"(<function __main__.<lambda>(A, B, E, L, M, N, P, R, U, Y)>, 'ABELMNPRUY')" "(<function __main__.<lambda>(A, B, E, L, M, N, P, R, U, Y)>, 'ABELMNPRUY')"
] ]
}, },
"execution_count": 6, "execution_count": 7,
"metadata": {}, "metadata": {},
"output_type": "execute_result" "output_type": "execute_result"
} }
@ -216,9 +243,16 @@
"compile_formula(\"NUM + BER = PLAY\", verbose=True)" "compile_formula(\"NUM + BER = PLAY\", verbose=True)"
] ]
}, },
{
"cell_type": "markdown",
"metadata": {},
"source": [
"Now we're ready for the faster version of `solve`:"
]
},
{ {
"cell_type": "code", "cell_type": "code",
"execution_count": 7, "execution_count": 8,
"metadata": {}, "metadata": {},
"outputs": [], "outputs": [],
"source": [ "source": [
@ -229,14 +263,14 @@
" for digits in itertools.permutations((1,2,3,4,5,6,7,8,9,0), len(letters)):\n", " for digits in itertools.permutations((1,2,3,4,5,6,7,8,9,0), len(letters)):\n",
" try:\n", " try:\n",
" if fn(*digits):\n", " if fn(*digits):\n",
" yield formula.translate(str.maketrans(letters, ''.join(map(str, digits))))\n", " yield formula.translate(str.maketrans(letters, cat(map(str, digits))))\n",
" except ArithmeticError: \n", " except ArithmeticError: \n",
" pass" " pass"
] ]
}, },
{ {
"cell_type": "code", "cell_type": "code",
"execution_count": 8, "execution_count": 9,
"metadata": {}, "metadata": {},
"outputs": [ "outputs": [
{ {
@ -245,7 +279,7 @@
"'587 + 439 = 1026'" "'587 + 439 = 1026'"
] ]
}, },
"execution_count": 8, "execution_count": 9,
"metadata": {}, "metadata": {},
"output_type": "execute_result" "output_type": "execute_result"
} }
@ -256,15 +290,15 @@
}, },
{ {
"cell_type": "code", "cell_type": "code",
"execution_count": 9, "execution_count": 10,
"metadata": {}, "metadata": {},
"outputs": [ "outputs": [
{ {
"name": "stdout", "name": "stdout",
"output_type": "stream", "output_type": "stream",
"text": [ "text": [
"CPU times: user 1.3 s, sys: 3.96 ms, total: 1.3 s\n", "CPU times: user 1.32 s, sys: 8.41 ms, total: 1.33 s\n",
"Wall time: 1.3 s\n" "Wall time: 1.33 s\n"
] ]
}, },
{ {
@ -273,7 +307,7 @@
"96" "96"
] ]
}, },
"execution_count": 9, "execution_count": 10,
"metadata": {}, "metadata": {},
"output_type": "execute_result" "output_type": "execute_result"
} }
@ -298,7 +332,7 @@
}, },
{ {
"cell_type": "code", "cell_type": "code",
"execution_count": 10, "execution_count": 11,
"metadata": {}, "metadata": {},
"outputs": [ "outputs": [
{ {
@ -372,8 +406,8 @@
"IN + ARCTIC + TERRAIN + AN + ANCIENT + EERIE + ICE + TRACT + I + ENTER + A + TRANCE = FLATIANA| 42 + 379549 + 5877342 + 32 + 3294825 + 88748 + 498 + 57395 + 4 + 82587 + 3 + 573298 = 10354323\n", "IN + ARCTIC + TERRAIN + AN + ANCIENT + EERIE + ICE + TRACT + I + ENTER + A + TRANCE = FLATIANA| 42 + 379549 + 5877342 + 32 + 3294825 + 88748 + 498 + 57395 + 4 + 82587 + 3 + 573298 = 10354323\n",
"ONE < TWO < THREE < SEVEN - THREE < THREE + TWO < THREE + THREE < SEVEN < SEVEN + ONE < THREE * THREE| 321 < 483 < 45711 < 91612 - 45711 < 45711 + 483 < 45711 + 45711 < 91612 < 91612 + 321 < 45711 * 45711\n", "ONE < TWO < THREE < SEVEN - THREE < THREE + TWO < THREE + THREE < SEVEN < SEVEN + ONE < THREE * THREE| 321 < 483 < 45711 < 91612 - 45711 < 45711 + 483 < 45711 + 45711 < 91612 < 91612 + 321 < 45711 * 45711\n",
"AN + ACCELERATING + INFERENTIAL + ENGINEERING + TALE + ELITE + GRANT + FEE + ET + CETERA = ARTIFICIAL + INTELLIGENCE| 59 + 577404251698 + 69342491650 + 49869442698 + 1504 + 40614 + 82591 + 344 + 41 + 741425 = 5216367650 + 691400684974\n", "AN + ACCELERATING + INFERENTIAL + ENGINEERING + TALE + ELITE + GRANT + FEE + ET + CETERA = ARTIFICIAL + INTELLIGENCE| 59 + 577404251698 + 69342491650 + 49869442698 + 1504 + 40614 + 82591 + 344 + 41 + 741425 = 5216367650 + 691400684974\n",
"CPU times: user 49.7 s, sys: 216 ms, total: 50 s\n", "CPU times: user 51.2 s, sys: 246 ms, total: 51.4 s\n",
"Wall time: 50.2 s\n" "Wall time: 51.7 s\n"
] ]
}, },
{ {
@ -382,7 +416,7 @@
"67" "67"
] ]
}, },
"execution_count": 10, "execution_count": 11,
"metadata": {}, "metadata": {},
"output_type": "execute_result" "output_type": "execute_result"
} }