Missing Square Puzzle: Neither Shape Is Really a Triangle

Two arrangements, one square short

Four pieces seem to fill a triangle 13 grid squares wide and 5 tall: right triangles with legs 8 and 3 and with legs 5 and 2, and L-shapes of 7 and 8 squares. Figure 1 shows them with every corner on a grid point.

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Now slide the same four pieces into a second layout, swapping the two triangles and moving the L-shapes without turning anything. The outline looks the same. Yet one grid square is left empty.

Illustration

A dissection puzzle is one set of pieces, cut so it rebuilds more than one shape. A true one keeps the area exactly, like the triangle that becomes a square in Haberdasher's Puzzle: Four Hinged Pieces, Triangle to Square; this one only seems to. Wolfram MathWorld files tricks like it under dissection fallacies: two figures that seem to use the same pieces but differ in area.

Find where the square went before you open the solution.

Show the solution

Lay a ruler on the long edge

Lay a ruler from the bottom-left to the top-right corner of either figure, and the long edge does not quite follow it. In the missing square puzzle, the large triangle climbs 3 squares for every 8 across (a slope of 3/8, or 0.375), while the small triangle climbs 2 for every 5 (2/5, or 0.4). The straight line climbs 5 over 13, about 0.385, between the two.

So the edge bends where the two triangles meet: in the first layout that corner is at grid point (8, 3), 1/13 of a square below the ruler (measured straight down), and the edge sags inward. In the second layout the small triangle comes first, and its corner at (5, 2) sits 1/13 of a square above the ruler, so the edge bulges outward. Each bend turns the edge by about 1.2°.

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Neither figure is a triangle, then. The first is a four-sided shape with a slight dent, and the second has a slight bulge.

Add up the areas

The pieces cover 12 + 5 + 7 + 8 = 32 squares in both layouts; a true triangle 13 wide and 5 tall would hold 13 × 5 ÷ 2 = 32.5. The dented first figure is half a square short of that, so the pieces fill it exactly. The bulging second figure is half a square over, so its outline holds 33 squares and the pieces leave one empty.

Shape	Area in grid squares
Right triangle, legs 8 and 3	12
Right triangle, legs 5 and 2	5
L-shape of 7 squares	7
L-shape of 8 squares	8
All four pieces	32
Straight triangle, 13 by 5	32.5
First figure, edge sagging	32
Second figure, edge bulging, empty square included	33

Put the two bent edges together and they enclose a thin four-sided gap with corners at (0, 0), (8, 3), (13, 5) and (5, 2). The gap is never taller than 1/8 of a square. But it runs 13 squares long, and its area is exactly 1. That gap is the missing square, spread along the whole edge.

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The lengths 2, 3, 5, 8 and 13 are consecutive Fibonacci numbers, and 8 × 2 − 3 × 5 = 1 gives the gap's area.

The check behind these numbers, in exact fractions, confirms that the pieces never overlap and leave no hole but the empty square. It also tries all 80 ways to split the 5-by-3 block into connected pieces of 7 and 8 squares; only 4 refill the 8-by-2 block with one square spare when the pieces slide without turning. All 4 are pairs of L-shapes.

The CSV file lists the corner points and exact areas of every piece, both outlines and the gap: missing-square-pieces.csv (CSV; where this file comes from: source, version and license, JSON) Source data: Computed for this page from exact grid coordinates of the four pieces

The same trick turns 64 squares into 65

Cut an 8-by-8 chessboard square into two right triangles with legs 8 and 3 and two trapezoids with parallel sides 3 and 5. Lay the four pieces out as a rectangle 5 wide and 13 long, and it seems to hold 65 squares, one more than the 64 you started with.

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In the rectangle, the cut edges climb 3 over 8 and 2 over 5, the triangle puzzle's two slopes. In the chessboard version, the pieces leave the same thin gap along the diagonal, with area exactly 1. Wolfram MathWorld uses this 64-and-65 cut as an example of a dissection fallacy.

Where people get stuck

Where does the missing square go?

Into the long edge. Neither figure is a true triangle: the first edge sags at its bend and the second bulges, so the first outline holds 32 squares and the second holds 33. The extra square is a thin gap along the whole edge, never more than 1/8 of a square tall.

Why can't you see the bend?

Each edge turns by only about 1.2° where the two triangles meet, and that corner sits 1/13 of a square above or below the straight line. On graph paper with 5 mm squares, that is less than 0.4 mm. Drawn with 1 cm squares, the offset is about 0.77 mm, twice as wide.

Who invented the missing square puzzle?

No published source that we have been able to read names the inventor of this four-piece version. Wolfram MathWorld says the same name is sometimes used for the Curry triangle, a six-piece figure created by L. Vosburgh Lions to show an effect Paul Curry discovered. That entry cites a 1959 book by Martin Gardner.