Lorenz Attractor

Pizza Pi Day Puzzle, Problem 1

The John Conway / Pizza Hut problems for Pi Day¶

http://blog.pizzahut.com/flavor-news/national-pi-day-math-contest-problems-are-here-2/

Problem 1¶

I’m thinking of a ten-digit integer whose digits are all distinct. It happens that the number formed by the first n of them is divisible by n for each n from 1 to 10. What is my number?

In [1]:

import itertools

Initial Observations

• The 10th digit must be zero.
• The 5th digit must then be 5.
• The digits in positions 2,4,6,8 must be even
• The digits in positions 1,3,7,9 must be odd
• The digit in position 4 must be a 2 or 6

Secondary Observations

• The first digit can be anything
• The ninth digit can be anything
• Given all the above conditions, the only positions that need to be checked are n=3,6,7,8
• There will be 288 candidates to be checked (12 even permutations x 24 odd permutations)

In [2]:

def check_nth_slice(workstring,n):
    if int(workstring[:n])%n==0: 
        return(True)
    else:
        return(False)
    
for even in itertools.permutations(['2','4','6','8']):
    if (even[1]=='2' or even[1]=='6'):
        for odd in itertools.permutations(['1','3','7','9']):
            candidate=odd[0]+even[0]+odd[1]+even[1]+'5'+even[2]+odd[2]+even[3]+odd[3]+'0'
            if check_nth_slice(candidate,3) and check_nth_slice(candidate,6) and \
            check_nth_slice(candidate,7) and check_nth_slice(candidate,8):
                print(candidate)

3816547290