The Top-Brick Shortcut: Why 1, 3, 3, 1 Makes Pyramids Easy

Climb a four-row number pyramid the usual way and you add three rows of bricks to reach the top. There is a quicker road. If you know the bottom row, you can jump straight to the top. The trick is a pattern: 1, 3, 3, 1.

The slow way first

Here is a pyramid with bottom row 3, 5, 2, 6. We climb it brick by brick.

The shortcut

Each bottom brick reaches the top by a number of different roads. The outside bricks have one road each. The two in the middle each have three. The picture shows how many times each bottom brick is counted.

So the top of a four-row pyramid is the first bottom brick, plus three times the second, plus three times the third, plus the fourth. Check with ours:

• 3 × 1 = 3
• 5 × 3 = 15
• 2 × 3 = 6
• 6 × 1 = 6
• 3 + 15 + 6 + 6 = 30. The same top, in one go.

Why it works

Look at the middle bottom brick, 5. It is added into the left brick of the third row (3 + 5) and into the middle brick of the third row (5 + 2). Each of those goes into the second row, and the 5 inside them gets added again. The middle bricks get counted more often because more roads lead from them to the top. Three roads each, one road each for the sides: 1, 3, 3, 1.

A shortcut that finds missing bricks

The same pattern works backwards. Suppose the top is 30, the bottom row is 3, ?, 2, 6, and the second brick is missing.

• Step 1. The known bricks count: 3 × 1 + 2 × 3 + 6 × 1 = 3 + 6 + 6 = 15.
• Step 2. The top is 30, so the missing brick counted three times must make up 30 − 15 = 15.
• Step 3. 15 ÷ 3 = 5. The missing brick is 5, exactly the brick we had before.

Check it with the slow way if you like. Same answer, one step.

Taller pyramids

The pattern grows. A five-row pyramid counts its bottom bricks 1, 4, 6, 4, 1. A six-row pyramid uses 1, 5, 10, 10, 5, 1. These are the rows of Pascal's triangle, in which each number is the sum of the two above it, which is the same rule as our pyramids.

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