# Making Maths Count

## Maths Week Scotland – Dotty Rectangles

September 11, 2018 by John Swinney MSP 5 Comments

It’s day two of Maths Week Scotland 2018. Every day this week I’m sharing a new maths puzzle, so here’s the second challenge, and remember to come back for the solution and a new challenge tomorrow.

Good luck!

**Dotty Rectangles**

Here is a 6×3 rectangle drawn on dotty paper. There are 10 dots inside.

(a) How many dots would there be inside a 9×4 rectangle drawn on dotty paper?

(b) A rectangle with area 24 (square units) is drawn on dotty paper. How many dots could there be inside? (Note that there is more than one correct answer.)

(c) A rectangle is drawn on dotty paper so that its length is twice its width. Its perimeter is 54 (units). How many dots are there inside it?

**Solution to Dotty Rectangles**

Answers: (a) 24 (b) 0, 11, 14, 15 (c) 136

(a) A 9×4 rectangle would have 24 dots inside (arranged in an 8 by 3 pattern).

(b) The dimensions of the rectangle could be 24×1,12×2,8×3 or 6×4. So the possible numbers of dots inside are 0, 11, 14, 15.

(c) If the perimeter is 54, then length + width = 27. Sharing 27 so that one number is twice the other gives 18 and 9. So the length is 18 and the width is 9. Therefore, the number of dots inside is 17×8=136.

Tags: John Swinney, Maths Week

a) 24 dots

b) 6×4 rectangle- 8 dots

8×3 rectangle – 6 dots

c) 18×9 rectangle – 112 dots

That looks more like a 7×4 rectangle?

Thanks for your comment and for trying this puzzle. The rectangle is 6 squares by 3 squares (although you are right to point out that this means an extra dot on each side!). I hope this helps.

This is not a 6×3 rectangle. It’s a 7×4 dots rectangle.

confusing

but there would be 14 dots in a 9×4 rectangle.

you’re welcome

Miss Molina and regi class A6

Thanks for your comment and for trying this puzzle. It’s great that you have found different solutions depending on the size of the rectangle. The rectangle is 6 squares by 3 squares (although you are right to point out that this means an extra dot on each side!). Sorry for any confusion! I hope this helps and that you are enjoying Maths Week Scotland 2018.