### Pyramids

What are the missing numbers in the pyramids?

A little bit of algebra explains this 'magic'. Ask a friend to pick 3 consecutive numbers and to tell you a multiple of 3. Then ask them to add the four numbers and multiply by 67, and to tell you the last two digits of her answer. Now you can really amaze her by giving the whole answer and the three consecutive numbers used at the start.

### Score

There are exactly 3 ways to add 4 odd numbers to get 10. Find all the ways of adding 8 odd numbers to get 20. To be sure of getting all the solutions you will need to be systematic. What about a total of 15 with 6 odd numbers?

# Always the Same

##### Stage: 3 Challenge Level:

Thank you to Jimmy, Natalie and Lucy at West Flegg GM Middle School for their work on the problem "Always the Same". Natalie is not the first student to add together all the numbers 1 through 16 and take the average.

Thus replacing the ascending numbers with 8.5 in every cell and circling four cells gives a total of 34. Or as Natalie did, she realised that "you pick numbers from each column and row" and took the average between the sum of the four columns:

i.e. (28 + 32 + 36 + 40)/4 = 34

A good solution with this method came from Melanie and Rachel of Flegg High School.

A proof of this problem could be as follows.

Let the first number be a.

 Then when choosing numbers from rows and column that do not coincide we have: a + (a + 4) + (a + 8) + (a + 12) + 1 + 2 + 3 = 'Magic number' We add 1 because there is one number in the second column, 2 because there is one number in the third column and 3 because there is one number in the fourth column.

Hence:

4a + (4 + 8 + 12) + ( 1 + 2 + 3) = 34
i.e. 4a + 30 = 34
i.e a = 1
and the array is 1 through 16 as set.

But suppose the 'magic number' had been 62 then

4a + 30 = 62
i.e a = 8
and the array would have been 8 through 23.

Hope that the explanation above helps especially Josh at Russell Lower School to "work out where we went wrong".

We could have used a 5 by 5 array of ascending numbers!