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.
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?
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.
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
We could have used a 5 by 5 array of ascending