What’s the effect?
A 4×4 square grid is created on a piece of paper with numbers from 1-16 (see top left hand picture). Spectators choose 4 numbers at random from the grid and the total always equals 34.
What you need?
- Pen and paper
What’s the method?
There was one crucial detail left out from the description of the effect. When the first number is chosen and circled, the remaining numbers on that row and column are crossed out (see the top right picture).
There is now only a choice of 9 numbers for the second spectator to choose from. Again the choice is circled and the row and column is crossed out. For the third choice there are only 4 numbers remaining. The final choice isn’t a choice at all as there’s only one number that hasn’t been circled or crossed out. The total of the 4 chosen numbers is 34. Every time! (see the bottom right picture for one possible example)This can be repeated with different choices to show that it always totals 34.
There are a number of extensions that can be made for classroom investigations.
There is a general formula that can be set as a task for students to find. The total T is the grid size n multiplied by the central number of the grid. The central number is obvious when odd numbered grid sizes are constructed. However it’s easy to calculate by adding the first and last numbers (n^2) and dividing by two. Hence the term in the brackets of the general equation.
A further investigation could explore the links between this trick and Latin Squares.