### Consecutive Numbers

An investigation involving adding and subtracting sets of consecutive numbers. Lots to find out, lots to explore.

### Have You Got It?

Can you explain the strategy for winning this game with any target?

### Counting Factors

Is there an efficient way to work out how many factors a large number has?

# Roses and Carnations

##### Age 11 to 14 ShortChallenge Level

0 roses $\rightarrow$ £56 left - even so can be spent on carnations
1 rose $\rightarrow$ £53 left - odd so cannot be spent on carnations
2 roses $\rightarrow$ £50 left - even so can be spent on carnations
3 roses $\rightarrow$ odd amount left - not good
4 roses $\rightarrow$ even amount left - good
etc

Only even numbers of roses until the maximum possible number of roses
£56 $\div$ 3 = 18 remainder 2 (1 carnation)
Maximum 18 roses
10 even numbers (including 0)
10 possible bunches

0 carnations $\rightarrow$ £56 left but 56 is not a multiple of 3
1 carnation $\rightarrow$ £54 left $\rightarrow$ 54$\div$3 = 18 roses
2 carnations $\rightarrow$ £52 left but 52 is not a multiple of 3
3 carnations $\rightarrow$ £50 left but 50 is not a multiple of 3
4 carnations $\rightarrow$ £48 left $\rightarrow$ 48$\div$3 = 16 roses
etc

3 more carnations $\Rightarrow$ £6 less to spend on roses = 2 fewer roses
Every 3rd carnation gives a possible number of roses

Maximum number of carnations: 56$\div$2 = 28
1, 4, 7, 10, ..., 28 allowed
2 less than the 3 times table, up to 30
so there are 10 possible bunches (since 30 = 3$\times$10)

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