Problem 32 - Pandigital Products

Problem

This problem comes from Project Euler 32

Problem

We shall say that an $n$-digit number is pandigital if it makes use of all the digits 1 to $n$ exactly once; for example, the $5$-digit number, $15234$, is $1$ through $5$ pandigital.

The product $7254$ is unusual, as the identity, $39 \times 186 = 7254$, containing multiplicand, multiplier, and product is $1$ through $9$ pandigital.

Find the sum of all products whose multiplicand / multiplier / product identity can be written as a $1$ through $9$ pandigital.

HINT: Some products can be obtained in more than one way so be sure to only include it once in your sum.

Solution

The first thing to realize is that there are only two possibilities ✌️:

• 1-digit numbers × 4-digit numbers = 4-digit numbers
• 2-digit numbers × 3-digit numbers = 4-digit numbers

Any other combination would cause the multiplier × multiplicand to exceed the product. For example, 2-digit numbers × 4-digit numbers ≠ 3-digit numbers.

You’ll also want to avoid 4-digit numbers × 1-digit numbers and 3-digit numbers × 2-digit numbers, as that’s just duplicating the work done above. I never had to consider this issue with the way I solved the problem, though.

Code

# Project Euler: Problem 32
# Pandigital Products

from itertools import permutations

numbers = ['1', '2', '3', '4', '5', '6', '7', '8', '9']
perms = permutations(numbers)

sums = 0
memo = {}

for perm in perms:
    # 1 by 4
    a = perm[0]
    b = ''.join(perm[1:5])
    prod = int(a)*int(b)

    if prod < 9999:
        if prod not in memo:
            prod_str = ''.join(sorted(a + b + str(prod)))

            if prod_str == '123456789':
                memo[prod] = True
                sums += prod

    # 2 by 3
    a = ''.join(perm[:2])
    b = ''.join(perm[2:5])
    prod = int(a)*int(b)

    if prod < 9999:
        if prod not in memo:
            prod_str = ''.join(sorted(a + b + str(prod)))

            if prod_str == '123456789':
                memo[prod] = True
                sums += prod

print(sums)

The code does the following:

1. Create a list¹ of every permutation of 1 to 9
2. For every permutation, get the 1-digit × 4-digit product and check if we’ve already discovered it’s pandigital, and if not, check if it’s pandigital
3. Do the same thing for 2-digit × 3-digit products

[1] Python doesn’t actually create a list, but a generator, which is more efficient in this case

Example

We’re essentially doing something like this (in pseudocode):

permutation = ['3', '9', '1', '8', '6', '4', '2', '7', '5']

# 1 by 4
a = 3
b = 9186
prod = 3*9186 = 27558

is prod < 9999? No
    Continue

# 2 by 4
a = 39
b = 186
prod = 39*186 = 7254

is prod < 9999? Yes
    is prod stored? No.
        Combine and Sort 39, 186, 7245 => 123456789
            is 123456789 pandigital? Yes. Add to sum and store

Most of the code is string-to-number transformations and vice-versa, as sometimes working with strings can be easier 🤷.