快猫短视频

#10 Chairs in pairs

Set by Peter Rowlett

Abbie and Bryn are allocated places to sit at random in a row of four seats. What is the probability that they are put next to each other?

What if there are 20 seats?

On a different occasion, Abbie and her friend Ana have booked two adjacent spots on the front row, which is 22 seats long. Bryn and his friend Beth also have two adjacent seats among the 22. What is the probability the two groups are next to each other?

Solution next week

#9 Rectangulator

Solution

From the 7, we can go round a rectangle, with width or height 1 or 2, in either direction. Eight numbers are possible: 7854, 7821, 7964, 7931, 7458, 7469, 7128 and 7139. If you allow zero width/height, you can also have 7777, 7887, 7997, 7447 and 7117. For zero-height or width rectangles, the first two digits are the last two digits reversed. This will always be a multiple of 11, since it will be a sum of multiples of 110 and 1001, both multiples of 11. If we extend the rectangle, we move both numbers horizontally or vertically by the same distance. Instead of 7117, we could extend to the right and get 7128. This won鈥檛 affect whether it is a multiple of 11, as we are adding or taking away 11 per column we move. Likewise, extending a zero-height rectangle horizontally, we just add or subtract 33 per row we move.