## Tuesday, August 25, 2015

### LUCKY (magic) SQUARES

8/25/2015

This is an interesting performance you can do with some people.  You start off explaining that you will calculate their “Lucky Square”, but you need their help.  You will pick a number to start with, but after that they will pick the numbers, and decide whether or not to add or subtract.
Amazingly, their lucky square turns out to be a magic square (all the rows, all of the columns, and both diagonals sum to the same lucky number).  You explain that the square that they helped you create has some “strong magic” and they should carry it with them always.
After doing this trick a few times you will have the order that you will calculate their numbers memorized.  But for now you need to memorize the chart below:
 8 1 7 This is the position grid you will use.  You will write the first number in position 1, the second number in position two, the third number in position 3, etc.  When I refer to using the number in position 1, it means the top middle position.  If I refer to using a number in position 2 it means the bottom left position.  Position 3 means the middle of the right column. 5 6 3 2 9 4
First obtain the participant’s birthday, and write it down above your blank grid.  It does not matter if you write it down with month first, or day first (MM/DD/YY or DD/MM/YY).  You may choose to use a four digit year instead of a two digit year, and this may be more helpful to you if the two digit year is a small number (it will avoid having negative number is the square).  If they don’t want to tell you their birthday ask them what three numbers seem to be the luckiest for them, and to start the trick start with the largest.
Suppose their birthday is:
“3 – 14 – 81”
 81 If the year offers only a small number, such as the year 2003, go ahead and write down all four digits to avoid doing a subtraction that will lead to negative number.  If you are not sure, go ahead and write down the four digit year – it won’t hurt.  In this case, it I add or subtract it probably won’ be an issue.  Now go ahead and write the year number in position 1.  Tell your participant: “This is one of the most important numbers for your square.”

 81 Now ask the participant which number they want to use next (either the 3 or the 14), and ask if they want to add or subtract the chosen number.  Suppose they chose the 14, and want to subtract.  Then take the 81 and subtract 14 (which is 67), and then subtract 14 again (which is 53).  Write the 67 in position 2, and write the 53 in position 3.  And tell the participant that “These are the three most important numbers in your Lucky Magic Square.” 53 67

 81 73 Next, ask the participant what number they want to use next, and do they want to add or subtract.  Suppose they choose this time to use the three, and they want to add it.  Start with the 81 in position 1, then add 81 + 3 = 84, and then add it again.  84 + 3 = 87.  Write the 84 in position 4, and the 87 in position 5.  Next use the 67 in position 2, add 67 + 3 = 70, then add 70 + 3 = 73.  Write the 70 in position 6, and the 73 in position 7. 87 70 53 67 84

 56 81 73 Finally, we use the 53 in position 3, and add 53 + 3 (which is 56) and add 56 + 3 (which is 59).  Write the 56 in position 8, and the 59 in position 9. This completes the magic square, but you now have to explain why this square is so special, or so lucky.  “You are not going to believe this …”  Show them that each row has the same sum (in this case 210), and that each column has the same sum (in this case it is still 210), and that each main diagonal has the same sum (in this case 210 again). 87 70 53 67 59 84
Explain that not all Lucky Squares have this particular property (even though they do).  It is very magical and is therefore very lucky.  They need to make sure that they carry it with them at all times, and make sure they don’t lose it.  Give them a paper copy of their Lucky Square.
Now suppose that the next birthday you are asked to use is:
“8 – 23 – 14”
Please be very carefully preforming and explaining what you are doing at each step – because the participant is either very young or very old.
 2014 In this case it will be better to use a four digit year.  So write 2014 in position 1. Then we ask which number the participant wants to use next, and if they wish to add or subtract the number.  Suppose they say they want to add using the 8.  In position 2 we add 2014 + 8 (which is 2022), and in position 3 we add 8 again (which is 2030).  Don’t forget to explain that these are the three most important numbers in their lucky square. 2033 2022

 2014 Now we ask again which number do they want to use next, and do they want to add or subtract. Suppose they want to use the 23 next, and they want to add again.  Starting with 2014 in position 1, we add 23 (which is 2037) and write the answer in position 4, and then add 23 again (which is 2050) and write the answer in position 5. 2050 2030 2022 2037

 2053 2014 2068 Remember, we are still adding 23.  We take the number in position 2 and add 23 twice (which is 2045 and 2068) and write them in positions 5 and 6. Finally, we take the number in position 3 and add 23 twice (which is 2053 and 2076) and write these numbers in positions 8 and 9. The square is completed, and it has a magic constant of 6135.  Every row, every column, and both main diagonals sum to Tell them, “You are not going to believe this but ...” and show them the magical properties of their square. 2050 2045 2030 2022 2076 2037
Again, don’t forget to explain that they have a very special lucky square, you could even say magical since most other lucky squares don’t have this property.  Make sure you give them a written copy of their lucky square, and explain they should carry it with them always.

The material above was adapted from: “Mathematical Magic” by William Simon
Sources
“Mathematical Magic” by William Simon

David