*Date: 11/17/2018*

I have been interested in magic squares longer than I have been interested in magic. And I have been interested in magic for a long time. It was actually a close cousin of the magic square, the magic matrix as described by Martin Gardner in one of his mathemagic books, which got me into magic. But that is a story for another day.

In this article I want to explain two simple ways to construct an arbitrary 4x4 magic square.

6 | 11 | 0 | 13 |

1 | 12 | 7 | 10 |

15 | 2 | 9 | 4 |

8 | 5 | 14 | 3 |

Without going too much into the history and mathematics of magic squares, let me just say that there are several different 4x4 magic squares which you could substitute with the one shown here. This square is a bit unusual because it does not use the numbers 1 - 16 as usually the case but rather the numbers 0 - 15. This has a special reason to simplify calculations later on. The sum of the magic square is 30.

Convince yourself that this is indeed a magic square. Every row adds up to the same number 30. Every column, the two main diagonals, each of the four sub-squares, the center square, and the four corners, everything adds up to 30. You can find other combinations that add to 30. However this article is not meant as an exhaustive treatise on magic squares, but rather as a quick but solid introduction to perform an effect.

Ok, now that you have memorized this magic square, or recorded it somewhere secretly where you can copy it from, observe the following neat little property of our 4x4 magic square:

6 | 11 | 0 | 13 |

1 | 12 | 7 | 10 |

15 | 2 | 9 | 4 |

8 | 5 | 14 | 3 |

If you add any number X to the four numbers in the grey cells, the magic square remains a magic square and its sum becomes 30 + X. For example, add 5 to the four grey squares to get a magic square that sums to 35:

6 | 11 | 5 | 13 |

6 | 12 | 7 | 10 |

15 | 7 | 9 | 4 |

8 | 5 | 14 | 8 |

If you wanted to get a magic square summing to less than 30 you would have to introduce negative numbers. But it is usually better to subtly force a spectator to pick a number 30 or higher. You could say something to the effect of: "To make it more interesting pick a number higher than 30." This is all you need to know to write down a magic square adding up to any spectator selected number.

Of course, you will quickly see that this most simple method has a problem with high numbers. Say the spectator picks 154. Then your magic square would look like:

6 | 11 | 124 | 13 |

125 | 12 | 7 | 10 |

15 | 126 | 9 | 4 |

8 | 5 | 14 | 127 |

The four squares you change have three digit numbers and all the other squares which do not change have one or two digit numbers. This does not look beautiful and it communicates the method. An observant spectator could easily reconstruct the method you are using. As they say there is no free lunch. A simple method often comes with certain deficiencies. Don't give up, because there is a simple remedy that requires a bit more work, but it is not very difficult either.

Combining the easy method with what we have just learned gives us the formula 30 + Y * 4 + X for the sum of our magic square. We have two degrees of freedom. We can increase the value of each and every square and or we can increase the value of our 4 special squares shown in grey above.

I know, some of you will be reminded of ugly math classes, but let me break it down for you into simple and easy steps - a recipe to follow. Once the spectator names his chosen number do the following:

- Subtract 30 from it. (Shouldn't be too hard since it is a round number to subtract.)
- Divide your result by 4 (If that is too hard divide by 2 and then again by 2.). Ignore the remainder for now. The number you are getting is your number Y in the formula above. It is the number you will add to each and every of the 16 squares.
- Either you remember the remainder of your division by 4, or take your calculated number Y multiply by 4 and then subtract from the number you get after subtracting 30. This is your number X. Add this number to the four grey numbers on top of whatever you are already adding because of Y.

- Substract 30: 154 - 30 = 124
- Divide by 2: 124 / 2 = 62. Divide again by 2: 62 / 2 = 31. Y = 31.
- Since there is no remainder X = 0. All we need to do is add 31 to each and every square to get the square below.

37 | 42 | 31 | 44 |

32 | 43 | 38 | 41 |

46 | 33 | 40 | 35 |

39 | 36 | 45 | 34 |

And now all numbers are close together with little fear of giving away the method.

Here is another example where X is not zero, 157:

- Subtract 30: 157 - 30 = 127
- Divide by 2: 127 / 2 = 63 (ignore remainder). Divide again by 2: 63 / 2 = 31 (ignore remainder). Y = 31.
- 31 * 4 = 124. 127 - 124 = 3. X = 3. All we need to do is add 31 to each and every square and an additional 3 to the four special squares.

- If you have a hard time remembering the starting magic square on the very top, then you could prepare sheets of paper with a 4x4 grid where each square is marked with its number in light pencil in a corner of the square. The marking of course has to be light enough that it is not easily visible for the spectators.
- If you are performing this on stage with a blackboard, then you have several possibilities to mark your squares or have a cheat sheet right on the blackboard. You could write the numbers with pencil on the blackboard. The pencil markings are not visible for the spectators, but the performer on stage can clearly see them. Or you could have a small version of the square somewhere on the edge or frame of the blackboard.
- Or read how Annemann presents a magic square in The Book Without a Name. With a clever presentation he eliminates any memorization.
- Once you have calculated X and Y you can decide how to fill in the squares. Probably the easiest method is not to go from left to right or top to bottom, but rather to go from the 0-square to the 1-square, 2-square, etc. and fill in numbers in sequence. If X is zero then you start with Y in the 0-square and sequentially fill in the numbers. If X is not zero you have to make sure to treat the first four squares differently, because they are the grey squares which also need to have X added to them.

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