### 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

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

### Always the Same

Arrange the numbers 1 to 16 into a 4 by 4 array. Choose a number. Cross out the numbers on the same row and column. Repeat this process. Add up you four numbers. Why do they always add up to 34?

##### Stage: 3 Challenge Level:

We received a mass of good solutions to this problem. Well done to you all.

Here is a collection of examples from Harry, showing that you always end up with a 5. The bottom line offers an algebraic explanation:

 Number (n) Add 3 Doubled Add 4 Halved Subtract n Answer 2 5 10 14 7 5 5 34 37 74 78 39 5 5 309 312 624 628 314 5 5 -23 -20 -40 -36 -18 5 5 -47 -44 -88 -84 -42 5 5 n n + 3 2(n + 3) 2n + 10 n + 5 n + 5 - n 5

Hussein, from Wilson's School, included a puzzle of his own that always ends up with a 1:

Think of a number.
Take away 2.
Then multiply by 3.
Add the number you started with.
Divide it by 4.
Take away the number you started with.

Can you work out how this works?

Neerajan, also from Wilson's School, sent us this:

I made up one and this is how it goes:

Think of a number,
multiply by 25,
take away 5,
divide by 5,
divide by 5.

Did you get the same number you started with?

Can you work out how this works?

Connor, from King John School, also created a similar puzzle:

Pick a number.
Double it.
Halve it.
Take the number you started with.

Did you get 3?

Can you work out how this works?

India, from Downe House, sent us this set of instructions:

Think of a number
Multiply by 4