Pizza Hut Pi Day Challenge (Part 8)

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 be one of the following 10 numbers:

1 , 4 7 2 , 5 8 9 , 6 3 0,

7 , 4 1 2 , 5 8 9 , 6 3 0,

1 , 8 9 6 , 5 4 3 , 2 7 0,

9 , 8 1 6 , 5 4 3 , 2 7 0,

7 , 8 9 6 , 5 4 3 , 2 1 0,

9 , 8 7 6 , 5 4 3 , 2 1 0,

1 , 8 3 6 , 5 4 7 , 2 9 0,

3 , 8 1 6 , 5 4 7 , 2 9 0,

1 , 8 9 6 , 5 4 7 , 2 3 0,

9 , 8 1 6 , 5 4 7 , 2 3 0.

Up until now, I have used the divisibility rules to ensure that the property works for n = 1, 2, 3, 4, 5, 6, 8, 9, 10. But I haven’t used n = 7 yet.

Step 10. The number formed by the first seven digits must be a multiple of 7. There is a very complicated divisibility rule for checking to see if a number is a multiple of 7. However, at this point, it’s easiest to just divide by 7 and see what happens.

$1 , 4 7 2 , 5 8 9 / 7 = 210,369.857\dots$: not a multiple of 7.

$7 , 4 1 2 , 5 8 9 / 7 = 1,058,941.285\dots$: not a multiple of 7.

$1 , 8 9 6 , 5 4 3 / 7 = 270,934.714\dots$: not a multiple of 7.

$9 , 8 1 6 , 5 4 3 / 7 = 1,402,363.285\dots$: not a multiple of 7.

$7 , 8 9 6 , 5 4 3 / 7 = 1,128,077.571\dots$: not a multiple of 7.

$9 , 8 7 6 , 5 4 3 / 7 = 1,410,934.714\dots$: not a multiple of 7.

$1 , 8 3 6 , 5 4 7 = 262,363.857\dots$: not a multiple of 7.

$3 , 8 1 6 , 5 4 7 / 7 = 545,221$: a multiple of 7!!!

$1 , 8 9 6 , 5 4 7 / 7 = 270,935.285\dots$: not a multiple of 7.

$9 , 8 1 6 , 5 4 7 / 7 = 1,402,363.857\dots$: not a multiple of 7.

So, by inspection, only one of these works, yielding the answer to the puzzle:

3,816,547,290.