Block Arrangement Problem
Arrange four blocks to satisfy specified contact counts. A contact must cover some area of a flat face; edges and corners alone do not count. The puzzle tests spatial reasoning and careful checking of constraints.
THE IDEA
Problem
Take four blocks (these may be matchboxes, books, cereal or detergent packets). The problem is to arrange them in certain specified ways. These ways are specified by how the blocks come to touch each other in the arrangement. For two blocks to be regarded as touching, any part of any flat surface must be in contact — a corner or an edge does not count.
The specified arrangements are as follows:
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Arrange the blocks so that each block is touching two others.
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Arrange the blocks so that one block is touching one other, one block is touching two others, and one block is touching three others.
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Arrange the blocks so that each block is touching three others.
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Arrange the blocks so that each block is touching one other.
Working with the Constraints
Contact means that some area of one flat face meets some area of another. The whole face need not be covered. Blocks meeting only along an edge or at a corner do not count as touching.
Try the arrangements with physical blocks and count the contacts for each block separately. Distinguish contacts within a layer from contacts between layers. For the all-to-all arrangement, two lower blocks can meet side by side while two upper blocks cross their seam. Each upper block must overlap both lower blocks, and the upper blocks must meet each other along a face.
The final condition does not require the arrangement to be connected. Two separated pairs can satisfy it. These constructions show why a precise definition and a physical check are more useful than assuming a configuration is impossible.
When to use it
Classroom exercise in psychology to demonstrate spatial reasoning and problem-solving heuristics.
Team-building activity where groups compete to find all solutions within a time limit.
Illustration of how precise definitions (e.g., 'touching') affect the solution space.
Warm-up puzzle for design thinking or engineering constraint analysis.
FROM THEORY TO EVERYDAY LIFE
In practice
Arrangement for Condition 1 (each block touches two others)
Place four identical blocks flat on a table in a square of two rows and two columns, with neighboring faces flush. Each block touches the block beside it and the block in the other row. Diagonal blocks meet only along an edge and do not count as touching.
Arrangement for Condition 2 (one block touches three, one touches two, one touches one)
Place two blocks side by side, then lay a third across their seam so that it touches both. Put the fourth on top of the third, clear of the lower pair. Each lower block touches its neighbor and the third block; the third touches all the others; the top block touches only the third.
A MOMENT TO REFLECT
Put it to work.
Why does contact over part of a face count, while contact only at an edge or corner does not? How would changing that rule affect the possible arrangements?
Try to solve arrangement #3 (each block touching three others). Do you think it's possible? Why or why not?
What mental strategies (e.g., drawing, physical manipulation, trial and error) did you use or would you recommend for this problem?
Often confused with
CONTINUE THE CONVERSATION
Source texts
- Lateral Thinking A Textbook of Creativity (Edward de Bono)