Puzzles

The digits 1 to 7 will each appear the same number of times as each other in each position of the number, so each digit of the mean will be the mean of the digits 1 to 7. Therefore the mean is 444.

The across number can be 138 (and the down number 234 or 432).

All the integers whose digits are all non-zero and add up to 9 can be made by taking a list of 9 ones, and between each pair deciding whether to move on a digit or not. For example, 1111|111|11 (where | means move on) would represent the number 432; and 1|1|1111111 would represent the number 117.

There are 8 gaps between 9s, so there are 2⁸ = 256 numbers that can be made.

Why are there not 512 positive integers whose digits are all non-zero and add up to 10?

The only three-digit number that is equal to the product of a square number and the same square number with its digits reversed is 16×61 = 976.

We can look at the first few powers and look for a pattern:

If n is even (and > 2), the last three digits of 15ⁿ are 625.

Holly's sum is odd, so she must have removed the units digit and so her calculation was:

\(a\) and \(b\) cannot both be 0, so the must sum in the tens column must've caused a carry into the hundreds column. This means that \(a\) must be 2, and the calculation is:

Two single digits cannot add to 19, so there can't be a carry from the units column into the tens column. This means that \(b\) is 8:

We can see now that \(c\) was 1, so Holly's three digt number was 281.

We can look for a pattern as we increate the number of 9s that make up n:

If n has k digits, then n³ is k-1 9s, followed by a 7, followed by k-1 0s, followed by a 2, followed by k 9s, The sum of all these digits will by 18k.

Hence, the answer is 18×55 = 990.

We can look at how many \(n\)-digit number there are for small values of \(n\) and look for a pattern:

These are the triangle numbers, and there are 231 20-digit numbers.

Why is the pattern the triangle numbers?