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Summing Consecutive Numbers

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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?

Your Number Is...

Stage: 3 Challenge Level: Challenge Level:1

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.
Then add 6.
Add the number you started with.
Divide it by 4.
Add 1.
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,
add 1,
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.
Add 6.
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
Add 2
Halve it
Add 7
Divide it by 2
Take away the number you first thought of.

The answer should always end up with 4.  
Can you work out how this works?