By Samuel Gugliotta
For Variety

IN my long and varied teaching career, I have always found that using a little recreational mathematics at the start of a lesson is a good way to get the students interested in the subject and to counter the apathy teachers are often are confronted with. So in this lesson I offer a few tricks for interested colleagues.
A few simple “magic” tricks based on mathematical principles will be my subject. Martin Gardner’s book, “Mathematics Magic and Mystery”’ and William Simon’s, “Mathematical Magic” are two classic introductions to the field, both published by Dover. The books are easy to follow, and you don’t have to be an expert magician or mathematician to learn a few exercises that may be fun and surprising for the students.
One category of tricks is based on standard dice. If you look at a standard die you will notice that the pips (the “dots”) on opposite faces always sum to 7. Thus 2 is opposite to 5, 3 to 4, 1 to 6. Now, being the “magician” of the class, you give a student (or group of students) 3 standard dice and ask her to roll them. Your back is turned when you do this, so you do not see the dice. But you ask the student to add the numbers on the top faces. She adds them, but keeps the number to herself.
(For example, I just rolled three dice, getting two 4’s and a 5. So the sum is 13. I note this sum, but I don’t tell you, the magician.)
Next, you tell the student to pick up one of the three die and add the number on its bottom to the total.
(In the present example, I picked up one the die which was showing 4, with has a 3 on the bottom, so I add this 3 to the total, giving me 16. I still keep this number to myself, and do not tell you, the magician.)
Next you tell the student to roll the die that she picked up again, and add the number showing to the previous total. (For example, I rolled the die with the four and got a two. Thus my new total is 18. I still do not tell you this total.)
You tell the student to leave the dice as they are, and finally you turn around. At this point, you glance at the numbers showing on the dice, and immediately tell the student what her total is. How do you do this?
Well you simply add up the numbers showing and add 7 to the result. (In the present example, there is a 4, 5, and 2 showing, adding to 11, and 11 + 7 = 18.
This works every time and is really based on simple mathematics. When you roll three dice you will get three numbers. Call them A, B, and C. So A + B + C is the initial total any time you roll three dice. Now if you take any of the dice, say the one with A showing, and add the number on the bottom of that dice, which is 7 - A, you will have A + B + C + 7 - A, which leaves you with B + C + 7. Now if A is thrown again, giving say a new number, D, the new total will be D + B + C + 7. Since, when you turn around, you see the D, B, and C, you need only add 7 to get the final result.
Another Dover book, “Amusements in Mathematics” by H. D. Dudeney, gives another trick based on throwing three dice. Again you ask your student to throw the dice without you seeing the result.
(For example, I just threw three dice and got a 6, 2, 2. )
Now you ask the student to pick one of the dice, and call this the first one. You ask her to multiply the points on the die by 2 and to add 5 to the result. (For example, I chose the 6, and 6 x 2 + 5 = 17.)
You then ask her to multiply the result (17) by 5 and add those points to the second die. So if my second die is the 2, I will now have 5 x 17 + 2 = 87.
Finally, you tell her to multiply that result (87) by 10, and add the total to the third die. In this case the result would be 87 x 10 + 2 = 872. You then ask her to tell you the result. Then once you hear the number 872, you can immediately tell her that her original throw was 6, 2, 2.
How can you do this? Well you simply subtract 250 from 872 (872 - 250 = 622). Again, this trick will work every time. If the original throw was A, B, C, we may express the process as 10(5(2A + 5) + B) + C, or 100A + 10B + C + 250. So for any throw, you need only subtract 250 from the result of the process to get the original A, B, C.
As a final example of mathematical “magic” I will give you a morsel from the category of calendar magic. Take any month of any year, and ask your student to block out a square of 9 dates. For example, for April 2007, I have blocked out thde dates 10, 11, 12, (first row), 17, 18, 19 (second row), and 24, 25, 26 (third row). Now you ask the student to give you the sum of any two dates diagonally opposite each other. In this example, I would give you 24 + 12 or 10 + 26 = 36. (The diagonally opposite numbers always sum to the same number.)
Now to determine the center number of this block, you simply divide by 2, giving 18. This is the date in the fifth position. To find the dates in the 4th and 6th position, to complete the row, you simply subtract and add 1, respectively. To find the other dates, you simply add or subtract 7 from the numbers in the central row.
A variation of this trick is to be able to give the sum of the block of nine numbers immediately upon being given the lowest number in the series. In the present example, you will be given the number 10. Now simply add 8 to that number and multiply the result by 9. (10 + 8) x 9 = 162. And this is the sum of the entire block of nine numbers.
The world of math and magic is a vast mindscape of entertainment. It’s logical, low cost, full of surprises and high returns.
Puzzles
1. Suppose you were planning to build a house. You were given the following costs: $1,100 for the roofer and the painter; $1,700 for the painter and the plumber;
$1,100 for the plumber and the electrician; $3,300 for the electrician and the carpenter; $5,300 for the carpenter and the mason; $3,200 for the mason and the painter. So what does each person charge for their services?
2. The local Social agency was able to dispense some cash to needy persons each week. If there were five needy persons less, then the remainder would get two dollars more each. However, at the next meeting there were four more needy persons than originally. As a result each person got one dollar less. If the same amount is distributed each week, how much is the amount, and how many needy persons were there?
3. A target had six concentric rings. Starting from the center bull’s eye, the points for each ring was 40, 39, 24, 23, 17, 16. How many arrows would be needed to score exactly 100 on the target?
Answers to Last Week’s Puzzles
1. 1,680 is another curious number, since if you add one to it the result is 1681, which is the square of 41. Half of 1680 is 840, and if you add one to that, you get 841, which is the square of 29.
2. 13458/6729 = 2, 17469/5823 = 3, 15768/3942 = 4, 13485/2697 = 5, 17658/2943 = 6, 16758/2394 = 7, 25496/3187 = 8, 57429/6381 = 9.
3. 42 x 138 = 5796, 18 x 297 = 5346, 28 x 157 = 4396, 27 x 198 = 5346, 39 x 186 = 7254, 48 x 159 = 7632.