# Find the perimeter (seemingly unsolvable problem)

It might seem that there is not enough information to solve this problem. But the fact is that there is enough information to find the perimeter.

• I will post the source after sometime. The source contains answers to this puzzle. Feb 4 at 19:20
• You might as well put it into a spoiler block then. Feb 6 at 10:46
• This is where the question has been taken from : twitter.com/Mathgarden/status/1618969850013155331 Please see the comments on the above tweet for solutions given by others. The author himself has also provided a solution which can be seen here : twitter.com/Mathgarden/status/1619743427440279553 Feb 6 at 11:08
• The intuitive challenge here is that you would not be able to calculate the area of this figure with the given information. Feb 6 at 19:15

Here's a non-visual solution which some may find more easy to understand than a visual solution:

Assume that the picture has north at the top. Suppose that we start at the northwest corner and begin walking east, and then continue following the figure until we get back to the northwest corner again.

The total distance that we walk east is 9 + 12 = 21, so the total distance that we walk west must be 21 as well. The total distance that we walk north is 15, so the total distance that we walk south must be 15 as well.

So the total distance that we walk overall is 21 + 21 + 15 + 15 = 72.

• This was the clearest for me Feb 6 at 2:50
• I solved it by doing the arithmetic of Oray's answer in my head, but I find this one the most elegant and (post-hoc) intuitively clear. Feb 6 at 12:16

To me the most visually intuitive solution is as follows:

First of all,

the vertical lengths on the right plainly add up to the vertical length on the left

Then take

that little unknown horizontal length (cyan) and move it to the bottom, moving the right sides to the right to match up

Now the two unknown horizontal lengths plainly add up to the sum of the two known horizontal lengths.

• This is the best answer of all +1 Feb 5 at 10:08
• Well, except that there is one more sentence missing. The one with the final number actually stated. Feb 5 at 20:09
• Regarding the horizontal sides, it is not clear that what you tell us to do will make your conclusion plain. Please could you clarify? Feb 7 at 9:19
• @RosieF Consider the sections that slide as a single piece--if the top moves by a certain distance, the bottom moves by the exact same distance. Feb 8 at 15:38

72

here is the solution;

sorry for my handwriting :D

• Your handwriting is better than mine! But then, I'm 63 with 45 years of heavy coffee drinking behind me. It is by caffeine alone I set my mind in motion. By the juice of the bean the thoughts acquire speed, the hands acquire a shake, the shake becomes a warning. It is by caffeine alone I set my mind in motion. Feb 5 at 20:11
• @BobaFit Dune quote?
– Oray
Feb 5 at 21:29
• Quote of a t-shirt based on the thing from Dune. Feb 5 at 22:35

An intuitive solution:

red is 15, blue 9, green 12

Perimeter is 2 x (red + blue + green) = 72.

In each of two steps rotate the highlighted bit of the perimeter by 180 degrees.

Variation:

The same principle presented more aesthetically but maybe not fully self-explanatory:

• Isn't this what I already posted? Feb 4 at 20:50
• I'll admit, my diagram is not as clear as yours. Feb 4 at 20:51
• @DanielMathias I hate to be blunt but I'm having a hard time understanding your diagram. I mean the whole point of a visual solution is clarity without technicality, isn't it? Feb 4 at 21:14
• I have edit my image, though my answer now seems redundant. +1 for better effort on your part. Feb 4 at 22:13
• Thank you for this different way of solving. I understood your solution. However, you have titled it, "an intuitive solution" but I am unable to figure out what the intuition is. To me, it just feels like we arranged the sides in a different way and were lucky that the figure became a rectangle. Can you please explain what the intuition is ? Thanks again Feb 5 at 18:09

A principled solution:

The perimeter length of

2 x sum of given lengths

follows from the following

Theorem:

Let P be a polygon with only right angles. Then the sum of all up facing sides equals the sum of all down facing sides and the sum of all left facing sides equals the sum of all right facing sides.

(We have WLOG turned the polygon so its sides face up, down, left and right.)

A technically 100% kosher (or halal if you prefer) proof is probably equally difficult as it is tedious. But, informally: This is certainly true for rectangles and remains true if we glue finitely many rectangles together which is all we need to do to build any such polygon.

• I really like this one. Feb 5 at 7:27

If we want to find the sum of all the vertical sides we have 15 and the other vertical sides on the right all add up to 15, giving us a vertical sum of 30. But if we want to find the horizontal sum, we will have to find the overlap between the 9 and 12 sides. Let the overlap be x. The bottom horizontal side will be 12 + 9 - x = 21 - x. However, the two sides that are labeled 12 and 9 add up to 21 and when including the overlapped side, we have an extra x, giving 21 + x. The -x and the +x cancel to give 42 + 30 = 72 for the perimeter.

An easy way to solve this is to just let the overlap be 0. Since the overlap could be any length, WLOG, we might as well let it be 0. Then the vertical sides sum up to 30 and the horizontal sides sum up to 21*2 = 42, giving 42 + 30 = 72.

• So, I saw your other answer and understood it. But, I don't understand this answer of yours. Specifically, I don't understand how we can let overlap be any length including 0 . Can you please explain what you are trying to say ? Feb 6 at 6:24
• This would have been my answer too. Maybe this solution needs the extra step of first asserting that the puzzle has a unique solution (the puzzle text states that "there is enough information"). If the solution is unique, and we have no information of the size of the overlap, then the perimeter must be independent of the size of the overlap, and we can set it to any value between 0 and 9, including the limit of the overlap approaching 0. Feb 6 at 8:21
• Given that there is a solution, then "overlap could be any length" and 0 makes it easy. IMO, the quick and most insightful way to solve this. Feb 7 at 16:52

Yet another proof without words:

A visual solution, without so much math:

Perimeter: (15+9+12) x 2 = 72

Edited to add colors to indicate the portions of the perimeter that are being translated. My original image was confusing, and loopy walt later presented a similar solution with a much clearer diagram.

• That's much clearer. And I wouldn't say it's the same. As the mathematical problem is very simple, all solutions will look similar in some way. But for the same reason all the value we can add is stylistic if you wish. So, small differences matter. Feb 4 at 22:26
• I cannot understand your answer. Firstly, how did you figure out that the red rectangle will have 2 sides of length 12 each ? Secondly, how does knowing that the red rectangle has 2 sides of length 12, help us figure out the perimeter of the original figure ? Feb 5 at 17:05
• How do you justify that the right hand side all lines up? Feb 7 at 14:18

assign x to the unlabeled horizontal length in the middle

assign y to two of the unlabeled vertical lengths, and (15 - 2y) to the other

starting in the lower-left corner and going up, walk the perimeter:

15 + 9 + y + x + y + 12 + (15 - 2y) + (12 - x + 9) =

72 + x - x + 2y - 2y =

72


I made a GeoGebra graph to show why the length of the cut-in doesn't matter. There is a slider for the length of the cut-in that can be modified to any value between 0 and 9 (the length of the top edge).

As others have noted, the perimeter is always 72.

A non-visual approach:

However long the two unlabeled horizontal sides are, their sum must always be the same: the shorter the short segment is, the longer the long segment will be, and vice versa. If we set the length of the shorter segment to zero, we see that this sum must be 21 (the sum of 9 and 12). Therefore, the perimeter is 2 x (15 + 21) or 72.

adding all the sides now, 9+15+(9)+12+(-4.5)+15+(-9)+12+(4.5)+4.5+4.5 =9+15+12+15+12+4.5+4.5 =9+9+12+12+15+15 =2(9+12+15) (we could try a different geometric approach from this conclusion too) =72