# Interesting geometry questions.

Scroll down to the second image to see both questions.

These questions were supposedly given to 10 year old students and baffled their college educated parents.

A lot of people have been saying it’s unsolvable but it’s not. Many are saying that the answers are 44 for the first question and also 44 for the second.The second problem is ambiguous and the answer assumes that the 2cm label refers to the vertical line to its left. It seems to me that this must be true or it wouldn’t be solvable.

I couldn’t understand how these two answers could be correct but after several minutes I understood that they are in fact the correct answers.

I’ll add that I at first thought that it depended on the drawings being in correct proportion. In other words, it could only be solved for the first drawing by physically measuring the image to determine the size of the small box in relation to the larger box and similarly for the second. This is wrong. The solution does not require measuring the images or that the drawings be in correct proportion. It took me several minutes to realize this.

I would hope that the principles involved here had been taught in class, otherwise it seems a little advanced for 10 year olds.

Yeah, it took me a few seconds to puzzle it out, too; it seems like there’s not enough information. Then I realized “duh, if the left side of the figure is 12cm, then the two segments that make up the right side of the figure must add up to 12cm”. I don’t know how long either segment is, and I don’t have to; just need to know that they add up to 12.

That analysis only works if the figures are made up of right angles. I looked up “rectilinear” and it said it only referred to straight lines; however searches for “rectilinear shape” or “rectilinear polygon” specify right angles as well.

As for the “larger box” in each question, I’m not sure that’s supposed to be part of the problem. I think it’s just the box where the student is supposed to write the answer.

When I referred to the “larger box” I wasn’t talking about the answer boxes on the right. I was referring to the image in the first question, which can be described as a larger box with a smaller box attached to the bottom right.

The solution to the top one:

Assuming the part jutting out is a square

Start at the top left corner:

The line from top to bottom is 12 so we have:

12

From bottom left corner to bottom right corner is 10

12 + 10

From the bottom right to the top right of the part jutting out is unknown. Call the extra bit L

12 + 10 + L

From top right corner of the jut back to the body of the shape is also L

12 + 10 + L + L

From the corner of the jut back into the body to the top right corner is 12 - L

12 + 10 + L + L + 12 - L

From top right corner to top left is 10 - L

12 + 10 + L + L + 12 - L + 10 - L

Adding the numbers together we get

44 + L + L - L - L

As you can see the Ls cancel each other out and we are left with 44.

That’s the complicated way to do it, but it gets the right answer. There’s a simpler more conceptual way to do it that will also make the second question easier.

Common sense, without any forumlas or theorems, told me the perimeter remains the same even if you cut out square corners. Like walking to the store, in a city with grid streets. It’s the same distance, whether you go all around the box, or take the intermediate streets in zigzag fashion.

The second one is only solvable if the 2-cm refers to the vertical lines. If the horizontal line is 2-cm, it is not solvable. You need to know backtrack distances.

Easy peasy. Are they trying to throw you off by drawing the 10cm length longer than the 12 cm length (in the first diagram)?

I think that what happens to a lot of people is that they get caught up in thinking about the missing chunk and end up thinking that the perimeter must be less than the perimeter of a 10 by 12 rectangle.That’s what happened to me at first.

The valid criticisms are that the shapes are drawn way off scale; and it’s not clear what measurement the 2cm in the second diagram is indicating (although of course only one is consistent with the problem being solvable). I suspect that there may have been a deliberate intention to draw them off scale to prevent measuring. If so, it might have been better to draw them just slightly off scale without being visually misleading, and to give an instruction to get the result without measuring.

All the father has succeeded in doing is showing the world that he and his friend are not too smart, A-levels and economics degree notwithstanding.

The scale isn’t important to the solution. The given measurements are all that’s necessary, but you’re right about the second one being ambiguous.

As long as you know the angles are all right angles, it’s easy to “move” the sides around so they form regular rectangles. Then it’s simple addition.

I also had a brief spell of confusion followed by a “duh!” moment, but I would expect the children have spent some class time working on similar problems and would be familiar with the presentation of the puzzles. In other words, having done this level of maths over 30 years ago is a hinderance rather than an advantage for me and other adults. Also, having learned more complicated maths since probably led to some initial overthinking on my part.

It’s not actually ambiguous. It’s not immediately obvious whether the 2 cm mark refers to the horizontal segment above or the vertical one to the left, but the problem is unsolvable if it refers to the horizontal segment. Since the problem does have an answer, we must assume that the mark refers to the vertical segment (in which case we get a 44 cm perimeter).

Right, but it’s ambiguous until you realize that.

It is certainly drawn ambiguously.

Sure, the ambiguity can be resolved by an assumption that the problem must have a solution, but why is that assumption warranted? It seems unlikely that 10-year-olds are being set problems with deliberate ambiguity that require such an assumption. If the kids are smart enough to understand such a sophisticated idea, then we might equally assume that they are set some problems where the correct answer is that there is no solution with the information given.

By far the most likely explanation is that it was just drawn poorly.

Yes, agreed. I would say the assumption is warranted because the question asks about “the perimeter” as compared to “a perimeter”, but that is a fairly sophisticated leap for a 10 year-old and I highly doubt the problem creator had that in mind.

I don’t understand the distinction you’re making between “the perimeter” and “a perimeter”.

Suppose the 2 cm referred to the horizontal line. Then, the possible perimeters could have ranged from (just over) 40 cm to (just under) 62 cm, assuming that the shape doesn’t self-intersect. If they had asked for a perimeter, anything in this range would have worked.

But asking for the perimeter implicitly assumes there’s only one possible perimeter regardless of the other dimensions, which is the case if the 2 cm refers to the vertical line.

The drawing is ambiguous but the assumption of a single answer resolves the ambiguity (though as said, it’s very unlikely the designer had this in mind).

Took me a bit, but then I don’t have a 1st class degree in economics. I’m only an engineer.

No matter what, it doesn’t take a college degree to know not to publicize one’s shortcomings on the internet. If you don’t get it, don’t complain publicly that it is too difficult until you find out if it really isn’t!