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tags: puzzle
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# Enlarging the Triangle Puzzle

Solving this math puzzle requires the same skills that are needed to debug code. In fact, as with a magician's trick, you can think about this puzzle as making us aware of a bug in your cognitive apparatus (assuming that you are, as I were, puzzled).

To be more specific, a hard-to-find bug in code puzzles us, because we are making implicit assumptions on the code (or the machine executing it). Often, the main difficulty consists in **making the implicit assumptions explicit**. This is also what happens in this math puzzle.

## The puzzle

The triangle on the left

 

can be rearranged to give the one on the right. How is that possible? [^footnote]

[^footnote]: Remember that two triangles with the same base and the same height, also have the same area.

Before we can start our detective work hunting for the hidden implicit assumption, we need to make explicit the assumptions we definitely know to be true. This can be done in at least two different ways.

Each will lead us to discover a different implicit assumption causing the puzzlement. In fact, we actually have two puzzles.

The first one is the more "magical" and the second one is already a logical reconstruction of the first.

## Version 1

Define the shapes on the left as follows:
- Split the base into 4 parts of lengths 3,2,2,3, respectively.
- Let the red triangles have height 7 and the blue triangles have height 5.

## Version 2

Define the shapes on the left as follows:

- Split the base into 4 parts of lengths 3,2,2,3, respectively.
- Construct a triangle of height 12 over the base.
- This defines the height of the red triangles (denoted b) and the height of the blue trianges (denoted a).
- Cut the middle rectangle into two grey L-shapes so that the height of the Ls is a.

## Task

In both versions of the puzzle, identify the implicit assumption that allows us to resolve the apparent contradiction. [^resolution]

[^resolution]: To be clear, the solution will then simply be to recognize that this implicit assumption we made was false.

<!--
After rearranging, we know:

- The new base is 5x2=10.
- The new height is a+b=12.

Because the two triangles have the same base and height, they have the same area.

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It logically follows that the black rectangle in the middle has no area, which means that $b/3 = a/2$, or
$$b/a = 3/2$$

Can we confirm that with a direct computation?

According the construction of the triangles, the angle $\alpha$ in the lower left satsifies $\tan(\alpha) = b/3$ but also $\tan(\alpha) = a/2,$ which, again, implies $b/a=3/2$.

-->