快猫短视频

#193 Flush tiling

Set by Alison Kiddle

Bonnie is refitting her bathroom and her plumber has brought round some very peculiar tiles to cover the floor. Her bathroom is a 5脳5 square and each tile is an L shape made up of three squares (see the diagram above).

Bonnie has realised it will be impossible to tile the whole bathroom without cutting tiles, as each tile covers three squares and 25 isn鈥檛 a multiple of 3. When she points this out, however, her plumber says it is OK, as they will need to leave one square of floor space free for the toilet.

鈥淲ell, clearly I don鈥檛 want the toilet in the middle of the room!鈥� she says.

In which squares could the toilet be placed?

Solution next week

#192 Hand over

Solution

After the merger, there are 10 board members.

When two people meet and shake hands, there is one handshake. With three people, there are three handshakes, four people have six and so on, and with N people there are N(N-1)/2 handshakes. These are the so-called triangle numbers: 0, 1, 3, 6, 10, 15, 21 etc. The number of handshakes was reduced by a third, so we are looking for a triangle number that is three times another number, and we know that N is less than 鈥渁 dozen or so鈥�. The only numbers that fit are N=10 (45 handshakes) and N=6 (15 handshakes). So, there are now 10 people on the board, of whom six refused to shake hands with each other.