## Tuesday, August 11, 2015

### Completing a 5 x 5 Pan-Magic Square (the easy way)

8/11/2015

8,112,015 is an arithmetic, composite, deficient, evil, odd, and wasteful number.  It is also a 2,704,006-gonal number.

COMPLETING A 5 x 5 PAN-MAGIC SQUARE:
This is another “Amazing Mathematical Feat” you can perform for your friends, family, students, or cohorts.
First, ask for a volunteer from the audience (which is easier if you have an audience of just one).  Ask them to choose a number between 1 and 100, and then ask them to pick a square from a 5 x 5 grid and write that number in the square that they choose.  For example: suppose they choose the number 14 and choose to place it in the middle square of the second row (as illustrated below).  Now explain that you are “going to place numbers in each of the 25 squares below, using all of the numbers from 1 to 25, in order to make a Magic Square.  All of this will be accomplished while leaving the chosen number 14 in the chosen square below.”
If you are working at a blackboard, white board, overhead projector, or similarly advance piece of educational technology, you should circle or otherwise indicate the beginning number and location.  Here, I have highlighted the square in yellow.
 14 15
Now here is what you need to know.
From 14, you will want to go up two squares, and then go right 1 square, and write the number 15.  However, that would take you outside the boxes.  Anytime your move takes you out the top of the box, go to the bottom box in that column and continue the move.  Likewise, if you move takes you out the right side of the box, go to the left most box in that row, and continue your move.  In this case, moving two block up, will place you just outside the boxes, so you move to the bottom box in the middle column – and then continue your move – one block to the right and write the number 15.
From 15, or from any number that is divisible by 5, instead of moving two boxes up and one to the right you will need to move two boxes to the right.  This will take you off the grid on the right, so move to the left most box on the bottom row.  And write the number 16.
For any number divisible by 5 remember to move two boxes to the right.
For any number not divisible by 5 remember to move two boxes up, and then one box to the right (like a Knights move in chess).
If you go off the top of the grid, continue at the bottom.  And if you go off the right side of the grid, continue on the left side.
Continue this until you get to a multiple of 25.  Your grid should look like this.

 18 24 21 14 20 17 23 25 19 16 22 15
Now go back to the 14 and start working backwards.
If the next number you are going to write is a multiple of 5, move left two blocks.  If you go off the left edge, move to the right most square in that row and continue counting you move.
If the next number you are going to write is not a multiple of 5, go down two blocks and to the left one block.  If you go off the bottom of the grid, move to the top box in that column, and continue counting.  If you go off the left side of the grid, move to the right most box in that row, and continue counting.
When you get to “1” your grid will be completed (like the grid below).

 12 5 18 6 24 8 21 14 2 20 4 17 10 23 11 25 13 1 19 7 16 9 22 15 3

Make sure you explain to your volunteer that this is a Pan-Magic Square: each of the four rows sum to (or “add up to”) 65, each of the four columns sum to 65, each of the two main diagonals and eight broken diagonal also sum to 65.  Your volunteer (and the audience) may be so amazed that they forget to applause.  Let them know that you understand and forgive them, that this is a common reaction to the amazing mathematical feat that you have performed – without a calculating device of any kind.  You can also show them that you have only 10 fingers, making these calculations impossible to do on your fingers alone.
If your volunteer chooses a number between 26 and 50, the magic constant will be 190.  For numbers between 51 and 75, the magic constant will be 215.  And for numbers between 76 and 100 the magic constant will be 440.
Suppose your next volunteer choose the number 77, and placed it in the second square in the fourth row (as shown below).  Then you follow the same rules.  Work forwards from 77 to 100, then backwards to 76.  Follow the same rules previous explained for multiples of 5, and for numbers that are not multiples of 5.  And the same procedures previously explained for going out of the grid on the top, bottom, left or right.
 76 94 82 100 88 97 90 78 91 84 93 81 99 87 80 89 77 95 83 96 85 98 86 79 92
If you are doing this in front of a classroom full of students there will always be a knucklehead (I’m sorry – I meant student) who will pick a number outside of the range you specified earlier – between 1 and 100.  Just explain that “The spirts are telling me that you are a non-believer.  You do not believe in following directions AND you do not believe that I can successfully complete this feat for the number you propose.  I, however, will accept your challenge.  And after I have amazed you with my mathematical magic, we will talk to your parents.”
Don’t worry – the process is the same.  Only the last two digits will change, the rest will remain the same in each block.  Suppose our beloved knuck … I mean beloved non-believer chose the number 1114 instead of 14   Then your completed square will look like this.  The have your beloved non-believer calculate the magic constant of the square as a homework assignment, but explain this is not for punishment, but because you know the non-believer would not believe you – he will only believe it if he himself performed the calculations.
 1112 1105 1118 1106 1124 1108 1121 1114 1102 1120 1104 1117 1110 1123 1111 1125 1113 1101 1119 1107 1116 1109 1122 1115 1103
If you are using a transparency on an overhead projector, and you really want to impress your audience – turn the grid 90 degrees and solve it that way.  (This is exceedingly difficult to do if you are using a chalkboard or a white board.)
You might also wish to impress your audience by doing the grid in Roman numerals, or Vulcan numerals (Klingons are not smart enough to do this – so don’t try Klingon):
The following numerals are written in the standard Vulcan script.  Sorry, they did not want to print out.  But you can go to the source and see what the Vulcan numbers look like:Source: http://www.languagesandnumbers.com/how-to-count-in-vulcan/en/vulcan/#numerals .

Have fun with this one.

David.

#### 1 comment:

1. A Diabolic Magic Square has the additional property of the magic sum along each of its broken diagonals. Curiously, we can construct a Diabolic Magic Square from a Graeco Latin Square! Read more about it on my blog: http://www.glennwestmore.com.au/category/latin-squares/.