5x9

Geometry Level 3

As shown below, I've cut a large, $5 \times 9$ rectangle into 10 smaller rectangles such that every one of them has integer side lengths.

Is it possible that the 10 small rectangles have all distinct areas?

Yes, it's possible No, it's not possible

1 solution

Áron Bán-Szabó
Aug 20, 2017

Consider ten rectangles, with distinct (integer) areas, such that the sum of the areas is the smallest. It is clear that it becomes the smallest, when the areas of the rectangles are $1, 2, 3, 4, 5, 6, 7, 8, 9, 10$ . The sum of the areas is equal to $1+2+3+4+5+6+7+8+9+10=55>45.$

Therefore it is not possible.

@Calvin Lin Isn't this puzzle is too easy? I mean that it is level 3, but I wouldn't write in the puzzle, that each rectangle has a distinct area 1 through 10. Maybe I would just write that they have distinct areas or changle 10 to a bigger number. For me that puzzle would be better.

Áron Bán-Szabó - 3 years, 9 months ago

What happened is that we've been trying to interpret and clarify your problem statements, because what you have written is often unclear. I believe you have a good idea in your head of the original problem, but it doesn't always come out well in the write-up. You can help us out by responding to the reports and clarifying your intent.

The original phrasing was:

Assume that we cut a $5\times 9$ rectangle into 10 smaller rectangles, such that each rectangle's sides are integers.
{{ image of rectangles labelled 1 to 10}} Is it possible, that any two rectangles have different areas?

In this case, it wasn't clear if the rectangles were meant to have areas 1 to 10, or were just labelled as such. Since it was about distinct areas, we went with the former interpretation.

Can you help edit the problem to the version that you are thinking of?

Calvin Lin Staff - 3 years, 9 months ago

5x9

1 solution

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