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 , 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.
where O is one of 1, 3, 7, and 9. (The last digit is 0 and not an odd number.) There are 96 possible answers left.
Step 8. The number formed by the first eight digits must be a multiple of 8. By the divisibility rules, this means that the number formed by the sixth, seventh, and eighth digits must be a multiple of 8.
For the first form, that means that 
For the second form, that means that 
We see that 
For the third form, that means that 
We see that 
For the fourth form, that means that 
In other words, we’re down to
O , 4 O 2 , 5 8 1 , 6 O 0,
O , 4 O 2 , 5 8 9 , 6 O 0,
O , 8 O 6 , 5 4 3 , 2 O 0,
O , 8 O 6 , 5 4 7 , 2 O 0.
For each of these, there are 3! = 6 ways of choosing the remaining odd digits. Since there are four forms, there are 4 x 6= 24 possible answers left.
In tomorrow’s post, I’ll cut this number down to 10.
Mean Green Math 
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