### Consecutive Numbers

An investigation involving adding and subtracting sets of consecutive numbers. Lots to find out, lots to explore.

### I'm Eight

Find a great variety of ways of asking questions which make 8.

### Calendar Capers

Choose any three by three square of dates on a calendar page. Circle any number on the top row, put a line through the other numbers that are in the same row and column as your circled number. Repeat this for a number of your choice from the second row. You should now have just one number left on the bottom row, circle it. Find the total for the three numbers circled. Compare this total with the number in the centre of the square. What do you find? Can you explain why this happens?

# Magic Squares for Special Occasions

##### Stage: 3 and 4

Published October 2000,February 2011.

In this article, Charlie Gilderdale records a meeting with the famous maths teach P.K. Srinivasan, who sadly passed away in 2005. You can read more about Srinivasan in his Wikipedia entry

On a recent visit to India I was fortunate to meet P.K.Srinivasan. He welcomed us on the 15 th August 2000 with a Magic Square which contained the date on the first row:

 15 8 20 00 14 11 13 5 2 18 1 22 12 6 9 16

He explained how this can be done for any date:

I will use this grid for reference purposes: (i) Place the special date in the first row
eg. Christmas Day
(ii)b+c = m+p
[there are many different possible values]
 a b c d e f g h i j k l m n o p
 25 12 20 00
 25 12 20 00 14 18

(iii) a+p = g+j
[ there are many different possible values]
(iv) m+d = f+k
[ there are many different possible values]
(v) b+n = g+k
[therefore n = 9]
 25 12 20 00 10 33 14 18
 25 12 20 00 3 10 33 11 14 18
 25 12 20 00 3 10 33 11 14 9 18

(vi) c+o = f+j
[therefore o = 16]
(vii) a+m = h+l
[there are many different possible values]
(viii) All rows, columns and diagonals must add up to the same total,
so e = 13 and i = 5.
 25 12 20 00 3 10 33 11 14 9 16 18
 25 12 20 00 3 10 31 33 11 8 14 9 16 18
 25 12 20 00 13 3 10 31 5 33 11 8 14 9 16 18
(check: a+d = n+o)
(check: d+p = e+i)

There are many different solutions, and the problem is trivial if we are allowed to repeat numbers; so the challenge is to complete the square without using any number more than once (but you will need to use negative numbers if the numbers in the top row add to less than 34).

Can you complete the Christmas Day Magic Square in a different way?

Can you complete a magic square with the date of your birthday in the top row?

There are some articles about magic squares on the NRICH website which you may like to see, Magic Squares from August 1998, its follow-up Magic Squares II and also Magic Sums and Products . A computer program to find magic squares shows how to program a computer to follow the method for finding magic squares described in this article. If this has whet your appetite there are some problems in the Archive which you might like to have a go at tackling (you can use the search box in the left hand margin to find them).

P.K.Srinivasan is the Curator-Director of the Ramanujan Museum and Maths Education Centre in Chennai (previously known as Madras). He is very welcoming to visitors and well worth contacting if you are ever in India (tel: 091-044-234 68 13).

 Charlie Gilderdale and P.K.Srinivasan working on some mathematics together.