School-Safe Puzzle Games

## The missing square: Where does it come from?

Here’s a real mind-bender:

If you re-arrange the pieces of the upper "triangle" to form the lower "triangle", a square goes missing. Can you explain? Please note that the pieces in both pictures are identical.

(picture from www.wikipedia.com)

### 28 Comments to “The missing square: Where does it come from?”

1. Tariq Nelson | Guest

The yellow section fits differently when put on the bottom, leaving a space between it and the green section

2. RK | Profile

Hi Tariq. Why do you think the dimensions of the bottom “triangle” (13×5) are the same as those of the top (13×5)?

3. G. Anderson | Guest

I suck at these kinds of problems but I’ll give it a try, because maybe what I say will help someone else.

I think it’s an optical illusion, having to do with the fact that Red and Blue triangles are switched. When the red triangle is put on top, it means the Blue triangle on the bottom does’t take up as much room leaving a gap that the yellow figure doesn’t fill.

4. RK | Profile

Thanks for stopping by G.

This is by no means is an easy problem; however, I think you’re on the right track when you say things aren’t as they appear-especially with the hypotnuse of the bottom “triangle” that the red and blue smaller triangles create…

5. Gaurav Dugar | Guest

the hypotenuse of the second triangle is not in straight line as in the first triangle.This creates an extra area taken by the empty square.

6. Gaurav Dugar | Guest

3/8 is the slope of hypo of bigger triangle.
2/5 is the slope of hypo of smaller triangle.
It’s like our eyes not able to distinguish between these slopes and they seem to be in parallel ,but they are not.
in actual these are not triangles,the are quadrilateral.

7. RK | Profile

Gaurav summarizes the essence of the solution quite nicely; this is famous puzzle, and there are many websites that manage to devote several pages to its explanation.

For those who still have a hard time understanding the answer, this site does the best job of explaining things:

8. Alfonso Macias | Guest

Gaurav is right and if you look closely the Top Triangle looks like it curves into the triangle while the Bottom Triangle curves out. Combining the two differences makes up for the extra square. Volumes are the same for only the colored areas. And since the slopes are different, neither hypotenuse is a straight line.

9. JOHN | Guest

While it’s true that the two triangles(red and blue) contain different slopes and this fact is often overlooked, it is not the simplest solution to the problem. When the triangles are placed as in the top illistration, they form two sides of an imaginary rectangle 3 x 5, for a total of 15 squares. When swithed, they form two sides of a rectangle 2 x 8, for a total of 16 squares. 16 – 15 = 1. Hence, the extra square. even someone with no understanding of geometry can solve the problem in this manner.

10. Lolease | Guest

I think it’s the blue, because even if the triangle moved they fit the squares the same.

11. Lord Gordo | Guest

The blue triangle is alot shorter than the big red triangle, therefor for the blue to line up equal with the red is if the yellow rectangle is moved back and leaves a empty box after the yellow rectangle.

12. Jean Paul | Guest

The different slopes is the key factor. The area of a 13×5 triangle is 32.5 but the sum of the area of the different pieces is only 32. Thus rearranging the pieces one can either compensate the missing .5 making in look compact (1st picture) or build a hole (2nd picture).

13. sophie | Guest

the top shape is quadrilateral not triangular

14. Abhijit | Guest

The space allotted to the portions colored yellow and green taken together increases by by one small square . This is due to differential ( small ) increase in the total area of the triangle after rearrangement of the different sectors i.e, red , blue green and yellow . However only minute obseravation lets us see the change in area covered by the different sectors .

15. Ashleigh | Guest

That was easy and i am only 10!!

16. RK | Profile

Ashleigh- what math classes have you taken at 10?!?

17. Ashleigh | Guest

I have only taken 1 math class per day but am the smartest in my math group or class

18. Tommy | Guest

I understood the Monty Hall problem when I was 8 or 9. This may be caused by the same thing as the Flynn Effect, or this may because I rock at math. I actually do it competitively. I ranked 33rd in the nation in MATHCOUNTS last year, though I should of done a lot better.

19. RK | Profile

Tommy- just curious, when did family/teachers realize you were so good at Math? Anyone in your family have a math/engineering educational background?

20. Tommy | Guest

I’ve been pretty good at math for almost as long I remember. My dad used to teach math in China. Now, he’s a bio statistician.

21. bilbao | Profile

Original source:
According to Martin Gardner, a New York city amateur magician Paul Curry invented this popular paradox in 1953

22. toxicasset | Profile

This is an easy square area problem….

It’s the resulting rectangular area you need to focus upon!
They are different to one another.
One is a 5 x 3 and the other is an 8 x 2. One has a square area of 15 and the other a square area of 16….

Simplezzzzzz

23. toxicasset | Profile

‘Hypo’ this ‘hypo’ that…..The area of the two triangles are EXACTLY the same you foolish people….

aaaaarrrggghhhh

24. mcmike | Profile

This has to do with Curry’s Paradox and the Fibonacci Sequence. The figure you are looking at is not actually a triangle! If you look at the two smaller triangles, one has a slope of 3/8 and one 2/5. If they have different slopes, then the hypotenuse is not actually a straight line segment, but rather two line segments. The area shown above is actually the same exact area you are seeing when the pieces are mixed up. Every missing square triangle like this has two triangles with slopes that are just ratios taken from the Fibonacci Sequence 1,1,2,3,5,8,13. Notice that 2/5 and 3/8 are both ratios of a(sub n)/a(sub n +2). Its quite neat!

25. fanfan | Profile

It is cunningly simple. The slopes of the blue and red triangles are different. The blue triangle has a steeper slope than the red. Hence in the upper diagram, the hypotenuse is slightly concave, ie., it is not a straight line. In the lower diagram, when the blue and red triangles are switched, the hypotenuse becomes convex, or bulged out. Therefore, the area of the bottom convex full triangle, including the missing square, is larger than the upper concave triangle. The difference in the areas, caused by the concavity and convexity of the hypotenuse, because of the difference in slopes of the blue and red triangles, IS THE EXTRA SPACE TAKEN BY THE LOWER DIAGRAM INTO WHICH THE MISSING SQUARE HAS DISAPPEARED.

26. alexanderabner | Profile

Well the hypotenuse slope of the right angled triangle is well hidden with different colours as a single slope,in fact it is two different slopes. In the triangle above the hypotenuse converges inwards and in the one below it converges outwards hence covering an extra area equal to a single square which obviously comes up when the yellow and green blocks are re-arranged.

27. Fawned Fridays | Sum of Sara | Guest