On March 14, 2016, Pizza Hut held a online math competition in honor of Pi Day, offering three questions posed by Princeton mathematician John H. Conway. As luck would have it, years ago, I had actually heard of the first question before from a colleague who had heard it from Conway himself:
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?
I really like this problem because it’s looks really tough but only requires knowledge of elementary-school arithmetic. So far in this series, I described why the solution must have one of the following four forms:
O , E O 2 , 5 8 O , E O 0,
O , E O 8 , 5 2 O , E O 0,
O , E O 4, 5 6 O , E O 0,
O , E O 6 , 5 4 O , E O 0.
where E is one of 2, 4, 6, and 8 (not already included) and O is one of 1, 3, 7, and 9. (The last digit is 0 and not an odd number.) There are 192 possible answers left.
Step 7. The number formed by the first four digits must be a multiple of 4. By the divisibility rules, this means that the number formed by the third and fourth digits must be a multiple of 4.
In other words, the two digit number 
also has to be a multiple of 4, which means that 
In other words,
O , E O 2 , 5 8 O , E O 0,
O , E O 6 , 5 4 O , E O 0.
At this point, I can include the remaining ways of choosing the even digits:
O , 4 O 2 , 5 8 O , 6 O 0,
O , 6 O 2 , 5 8 O , 4 O 0,
O , 2 O 6 , 5 4 O , 8 O 0.
O , 8 O 6 , 5 4 O , 2 O 0.
For each of these, there are 4! = 24 ways of choosing the remaining odd digits. Since there are four forms, there are 4 x x 24 = 96 possible answers left.
In tomorrow’s post, I’ll cut this number down to 24.
Mean Green Math 
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