Magic Square Fill-in Empty Cells Puzzles

Magic squares are one of the simplest forms of logic puzzles, and a great introduction to problem solving techniques beyond traditional arithmetic algorithms. Each square is divided into cells, and the rules require that the sum of any row, column or diagonal in the square be the same. Given a magic square with empty cells, your job is to solve the puzzle by supplying the missing numbers. 


The 3x3 magic squares on the attached puzzle worksheets are the least complex form of magic squares you can solve. These are normal versions (with numbers 1-9) that have a magic number of 15. All students should solve these puzzles but this should be the only challenged attempted by K-1st graders unless they work with their parents on the 4 x 4.


We also discussed how to calculate the magic number from just knowing the dimension of the square. For example, in a 3x3 magic square we have nine numbers. If we add the numbers 1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + 9 we can see that the first and last number (1 and 9) have a sum of 10 and the second and second to last number (2 and 8) have a sum of 10 and so on for four 10s with just 5 left over so 1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + 9 = 45. Since there are three rows, divide 45 by 3 and you get the magic number 15. 


In a 4 x 4 magic square we look at 1+ 16 = 17 and we know that we will have 8 pairs of 17. Using distributive property, 8(10 + 7) = 80 + 56 = 136. When we divide 136 by 4 = 34 or take half of 136 twice (68 and then 34), we generate the magic number of 34. The older students looked at a 10 x 10 magic square and saw the product of 101 x 50 = 5,050 and divided it by 10 to get 505 as the magic number.


The 4 x 4 fill-in the empty cells puzzle were very challenging since they very quickly exhausted the rows, columns or diagonals with already three numbers. When you have only two numbers in a row, column or diagonal, you have to look at two cells that will need a certain sum. If you cross out each used number and look at the remaining numbers you can see all of the pairs available. For example, if a row has two numbers with a sum of 20, you know that you need the other two cells in that row to have a sum of 14. Scanning the remaining numbers, if you have a 10 and 4 and those are the only two numbers that sum to 14, write both numbers in each empty cell in small font as a place holder. Inspecting intersecting columns or diagonals, you can see which number is required and which number can’t be used. 


The 5 x 5 puzzles are extremely hard (magic number 65) and should be attempted only after a mastery of the 4 x 4 puzzles.


My objective for this lesson is to teach children how to be logical thinkers with strategic methods of problem solving. Being able to see multiple approaches at the same time will build this strength. Chess is the ultimate multi-variable strategic thinking challenge.

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