Number pyramids

In a number pyramid, each number is the sum of the two numbers directly below it. Give the bottom row and the rest fills itself in: 4 and 9 give 13; 9 and 1 give 10; 13 and 10 give 23.

Going down is subtraction. If a box is 10 and one box below it is 4, the other must be 10 − 4 = 6. So a pyramid with some gaps can often be finished with a mix of adding and subtracting: with 10, 4 and 1 given, we get 10 − 4 = 6, then 4 − 1 = 3, then 6 − 3 = 3.

But sometimes the gaps are placed so that you cannot start. Take a pyramid with 60 at the top, 12 and 8 at the two ends of the bottom row, and three empty boxes. To add you need two known boxes below; to subtract you need a known box above and one below. Neither is there.

Here is the move that unlocks it: fill the empty boxes with letter-numbers. Call the left box of the middle row a, the right one b, and the middle of the bottom row c.

Now write the pyramid rule for each box:

• a + b = 60
• 12 + c = a
• c + 8 = b

Put the last two into the first: (12 + c) + (c + 8) = 60. That gives 20 + 2c = 60, so 2c = 40 and c = 20.

With c known, a = 12 + 20 = 32 and b = 20 + 8 = 28. Check: 32 + 28 = 60. Correct.

Is there a general link between the bottom row and the top? Start with the smallest pyramid: two boxes a and b below, one above. Its top is a + b. Build up from there and the pattern for bigger pyramids appears.