快猫短视频

#19 Angular arrangements

Set by Alison Kiddle

A regular polygon is a shape with at least three straight sides where all the sides are the same length and all the interior angles are equal.

Knowing that a full turn is 360掳 and the angles in a triangle add up to 180掳, can you use this diagram to work out the size of the interior angles in this regular pentagon? Hint: an interior angle is made up of two adjacent angles from the outer corners of the triangle.

Can you use the same idea to find a general way to calculate the interior angle of any regular polygon?

It is possible to arrange regular polygons around a point so they meet without leaving any gaps. Below is one such arrangement, using squares and equilateral triangles.

There are other ways to arrange regular polygons so they meet exactly at a point. How many can you find? What is the maximum number of sides a polygon in such an arrangement can have?

Solution next week

#18 The arithmetical two-step

Solution

From 42, we can subtract 2 and divide by 4, or divide by 6 and add 3. You can get to 10 in two steps for all numbers from 11 to 30. E.g. for 11, you undershoot and then come back by subtracting 2 and adding 1. Numbers from 12 to 28 can all reach 10 using two subtractions. For 29 you can subtract 9 and divide by 2. For 30 you can divide by 2 and subtract 5. For numbers above 30, if they aren鈥檛 a multiple of 10 you can subtract a number to make a multiple of 10, then divide to get 10. Multiples of 10 that are also a multiple of 20 can be divided by 2 then divided again to get 10, and similarly for multiples of 30. For 70, divide by 5 and then subtract 4. This leaves 50, which can reach 10 in one step but not two.