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?

Hot Dogs

Age 11 to 14 ShortChallenge Level

Using the average
60 hot dogs in 8 days is an average of 60$\div$8 = 7.5 hot-dogs per day.
7.5 will be in the middle of the numbers
7.5
4  5  6  7      8  9 10 11
4 hot-dogs on the first day

Sensible trial
6 hot-dogs on the first day, total 6+7+8+9+10+11+12+13=76 hot-dogs. Too many

5 hot-dogs on the first day, total 5+6+7+8+9+10+11+12=68 hot-dogs. Too many

4 hot-dogs on the first day, total 4+5+6+7+8+9+10+11=60 hot-dogs.
So she must have eaten 4 hot-dogs on the first day.

Using algebra
Suppose she ate $n$ hot-dogs on the first day. Then she ate $n+ 1$ hot-dogs on the second day, and $n+ 2$ hot-dogs on the third day, and so on until $n+ 7$ hot-dogs on the eighth day.

Altogether that is $n+1+n+2+n+3+...+n+7=8n+1+2+3+4+5+6+7=8n+28$ hot-dogs.

So $8n+28=60$, so $8n=32$, so $n=4$.

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