### Days and Dates

Investigate how you can work out what day of the week your birthday will be on next year, and the year after...

### Summing Consecutive Numbers

15 = 7 + 8 and 10 = 1 + 2 + 3 + 4. Can you say which numbers can be expressed as the sum of two or more consecutive integers?

### 1 Step 2 Step

Liam's house has a staircase with 12 steps. He can go down the steps one at a time or two at time. In how many different ways can Liam go down the 12 steps?

# Tiled Floor

##### Age 11 to 14 ShortChallenge Level

The sequence of all floors

Even floors: 2 purple tiles per row (total number even)
Odd floors: 1 more purple tile than the previous floor (total number odd)

109 purple tiles
108 = 2$\times$54 so floor length 54 has 108 purple tiles
$\therefore$ floor length 55 has 108 + 1 = 109 purple tiles
Floor length 55 has 55$^2$ = 3025 tiles

The sequence of odd-sided floors
Each odd-sided floor contains the last odd-sided floor in the middle, outlined in blue.

4 purple tiles are added each time
Side length 2 longer each time

109 = 1 + 4$\times$?
= 1 + 108
= 1 + 4$\times$27

So side length = 1 + 2$\times$27 = 55

$\therefore$ total number of tiles = 55$^2$ = 3025

Finding the number of rows
There are 2 purple tiles on each row,
Except for the middle row which has 1 purple tile (on odd-sided floors)

109 tiles = 1 purple tile in the middle
+ 108 purple tiles in pairs
= 1 purple tile in the middle
+ 54 pairs on 54 rows

$\therefore$ there are 1 + 54 = 55 rows
So there are 55$^2$ = 3025 tiles

You can find more short problems, arranged by curriculum topic, in our short problems collection.