4.8.2 Having Fun with Math—the Party Puzzle

The next activity is a number puzzle, which introduces some more ways of adding numbers quickly in your head. You will see how to turn it into a party trick to amuse your friends.

Activity symbol Activity: Party Puzzle

Write down the numbers 1 to 25 in order, in a 5×5 grid, so that the first row reads 1, 2, 3, 4, 5; the next row begins with 6, and so on.

Grid showing 5 rows, first row 1 to 5, second row 6 to 10, third row 11 to 15, fourth row 16 to 20, fifth row 21 to 25
Long description

Now choose five numbers from the grid as follows:

  • For the first number, choose any number from the grid. In our example below, we chose 17.
  • Now cross out the numbers in the same row and column as the chosen number.
This is a 5×5 grid. The first row from left to right contains the numbers 1, 2, 3, 4, and 5; the second row contains 6, 7, 8, 9, and 10; the third row contains 11, 12, 13, 14, and 15; the fourth row contains 16, 17, 18, 19, and 20 and the final row contains 21, 22, 23, 24, and 25. The number 17 is shaded and a line has been drawn through the fourth row and the second column.
Long description
  • Choose the second number from the remaining numbers on the grid—our choice is 24. Then, cross out the numbers in the same row and column as the second number.
This is a 5×5 grid. The first row from left to right contains the numbers 1, 2, 3, 4, and 5; the second row contains 6, 7, 8, 9, and 10; the third row contains 11, 12, 13, 14, and 15; the fourth row contains 16, 17, 18, 19, and 20 and the final row contains 21, 22, 23, 24, and 25. The number 17 is shaded and a line has been drawn through the fourth row and the second column, and the fifth row and the fourth column
Long description
  • Continue in this way until you have chosen five numbers. For example, let’s choose 17, 24, 15, 6, and 3. For your fifth number, there will be only one number left that’s not crossed out.
This is a 5×5 grid. The first row from left to right contains the numbers 1, 2, 3, 4, and 5; the second row contains 6, 7, 8, 9, and 10; the third row contains 11, 12, 13, 14, and 15; the fourth row contains 16, 17, 18, 19, and 20 and the final row contains 21, 22, 23, 24, and 25. The number 17 is shaded and lines have been drawn through every row and every column
Long description
  • Now add together the five selected numbers in your head or using pencil and paper. Check your answer with your calculator if you like. Here, the sum is sum with, 5 , summands 17 plus 24 plus 15 plus six plus three equals 65.
This is a 5×5 grid. The first row from left to right contains the numbers 1, 2, 3, 4, and 5; the second row contains 6, 7, 8, 9, and 10; the third row contains 11, 12, 13, 14, and 15; the fourth row contains 16, 17, 18, 19, and 20 and the final row contains 21, 22, 23, 24, and 25. The numbers 3, 6, 15, 17 and 24 have been shaded
Long description

Try the puzzle at least three more times, choosing different numbers each time. Work out the sums on paper or calculate in your head. What do you notice about the sums? Can you work out how to prove this?

Hint symbol

Comment

In the end, you will have selected exactly one number from each row, but none of these numbers appear in the same column. Find an easy set of five numbers from the grid that have this property.

Solution symbol

Answer

Whichever numbers you choose, you should find that the sum is always 65.

By specifying the way that the numbers must be chosen, exactly one number is chosen from each row and each column of the grid. It is then possible to prove that the sum will always be 65, without trying all the different choices of five numbers—that would be tedious!

In particular, you can choose the numbers on the diagonal from top-left to bottom-right because this gives one number in each row and column.

Grid showing 5 rows and 5 columns, first row is 1 to 5, second row is 6 to 10, third row is 11 to 15, fourth row is 16 to 20 and fifth row is 21 to 25
Long description

The sum is then sum with, 5 , summands one plus seven plus 13 plus 19 plus 25 equals 65. Finding the total of the diagonal elements is a quick way of working out the sum for this puzzle. Here, if we rearrange the order of the numbers so that we have sum with, 3 , summands open seven plus 13 close plus open one plus 19 close plus 25 and work out the sums in the parentheses first, we get sum with, 3 , summands 20 plus 20 plus 25, which is 65 as before.

Next is a general proof for why this trick works, if you’re interested in exploring the math a little more.

4.8.1 Codes

4.8.3 Understanding the Puzzle