1, 2, 3 Magic Square
1 out of 3
Arrange three 1s, three 2s and three 3s in this square so that every row, column and diagonal adds to the same total.
Problem
Arrange three 1s, three 2s and three 3s in this square so that every row, column and diagonal adds to the same total.
You might like to use this interactivity to do this problem.
This problem has been adapted from the book "Numbers in Your Head" by John Spooner, published by BEAM Education, which is now out of print.
Getting Started
What do one, two and three add to?
Student Solutions
This was a great problem! I don't think anybody spotted the fact that there was more than one solution, but we had the four different solutions sent to us individually.
Gaven from Forres Primary, Lucy and Richard who go to Flitcham V.A Primary School, and Ruth from Swanbourne House School all arranged their numbers like this:
Ruth says:
Whichever way you arrange the numbers, there will be one diagonal line with three numbers the same. So you have to work out which of the numbers, when multiplied by three, will come to the total you want in each of the rows.
This number is always the middle number of three when you choose three consecutive numbers.
Well noticed Ruth!
Katrina from Indian Ridge Middle School and Rachel from Histon and Impington Infant School both found another solution:
Rachel told us how she went about tackling the problem:
I added up 1, 2 and 3 to make 6, and tried to make each row add up to this. I started putting a 1 in the top right corner, and tried to put one of each number on each row or column. I nearly gave up when I saw all the 2's on the diagonal, but then I spotted that 2 plus 2 plus 2 also adds to 6.
Kim from Bottisham Village College and Natasha and Nathaniel both from Moorfield Junior School sent in a different solution again:
And the final solution was sent in by William who also goes to Moorfield Junior School:
Teachers' Resources
Using NRICH Tasks Richly describes ways in which teachers and learners can work with NRICH tasks in the classroom.
Why do this problem?
This problem requires only simple adding, but also persistence and logical thinking.
Key questions
Possible extension
Learners could try this activity with other sets of three consecutive numbers.
Possible support
Suggest using the interactivity or counters marked appropriately on a $3 \times 3$ square.