Depth · one concept from first principles · step 2 of 4

Find the rule of a growing block stack

Building blocks are stacked following a rule. How many blocks in all are needed to build the stack so that it is $7$ layers tall?

The block shapes grow from left to right, one figure after another.

Layer 1: $1$ block
Layer 2: $3$ blocks
Layer 3: $5$ blocks
Layer 4: $7$ blocks

Each figure is an L shape: one side rises one block taller while the bottom row also stretches one block longer. Each time a layer is added, the number of blocks increases by the same amount.

Operations

Your answer

blocks

How to solve

Strategy Look for a Pattern — The layer counts 1, 3, 5, 7 increase by 2 each time, so I extend that pattern to 7 layers, then add them up (building from the small known sums).

1STEP 1

Extend the layer pattern to 7 layers

Each layer adds 2, so extend 1, 3, 5, 7 to seven layers: the counts are 1, 3, 5, 7, 9, 11, 13 blocks.

1, 3, 5, 7, 9, 11, 13

2STEP 2

Add the blocks layer by layer

Add the seven counts step by step: 1+3=4, +5=9, +7=16, +9=25, +11=36, +13=49.

1+3+5+7+9+11+13 = 49

Answer

49 blocks

1+3+5+7+9+11+13 = 49

The sum of the first 7 odd numbers is 7 x 7 = 49, a known shortcut that confirms the layer-by-layer addition. 49 is a sensible total for a 7-layer growing stack.

Takeaway

Each layer grows by 2 blocks, and adding up the odd numbers 1+3+5+7+9+11+13 gives a neat 49!

Extend the layer pattern to 7 layers
Add the blocks layer by layer

Where next?

Level upsuggested

Perimeter of the nth figure by rule Sum-and-difference rules in a number grid

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8 practice problems

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13 reasoning problems grow out of this type — the bridge toward competition →

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▶ Practice — 8 problems

The same shape shows up later on the AMC, as Arithmetic Expression Decomposition, Geometric Series Share, Lattice / Grid Pattern from Small Cases. Why the shapes repeat →