A Calculator Trick

by Bill Price


This is a little math trick that I devised shortly after obtaining my first electronic calculator back in 1979.

Give your calculator to a friend. Tell him to key in any number between 1 and 900 (must be a positive integer), but don't let him tell you what it is. (Actually, this trick will work for numbers greater than 900, but on most calculators the resulting display will overflow. Hence the restriction). Tell him that he should try for a number difficult for you to guess, i.e., stay away from very small numbers like 1 or round numbers like 900. Try to make it any random three-digit number with mixed digits.  Now give him the following instructions:

  1. Tell him to multiply this number by 3.
  2. Then multiply the result by 7.
  3. Then multiply the result by 37.
  4. Then multiply the result by 11.
  5. Then multiply the result by 13.

(Note that these are all prime numbers containing the prime digits 1, 3, and 7. This is not really significant, but just sounds nice. If you like, you may point out this meaningless fact to your friend, to add to his stupefication.)

Ask him to show you the final result. From this final result, you will (with a little practice) instantly tell your friend what his original number was. For instance, from the result 93555462 you will be able to tell your amazed friend that the original number was 842!


This is how it is done.

First of all, note that 3 * 7 * 37 * 11 * 13 = 111111. So, in effect, you are having your friend multiply a number by 111111. Hopefully he took your advice and did not start with a simple one-digit number; otherwise the result arouses instant suspicion, and the trick is anything but impressive.

Depending on the beginning number, all results (for numbers 1 through 900) will appear in one of the following formats:

SSSSSS
SSSSSS0
SSSSSS00
HSSSSSF
HSSSSVF
HHSSSSVF
HHSSSVVF

where:

H = "head", consisting of zero, one, or two digits.
S = "stem", consisting of a series of easily identifiable REPEATING digits always located in the center.
V = "variable", consisting of zero, one, or two digits.
F = "final", always consisting of a single digit.
0 = the digit zero

If your friend ignores your advice and chooses a single-digit number, or a multiple of 10 or of 100, the result will of course be of the form SSSSSS, SSSSSS0, or SSSSSS00, respectively. In each of these cases, the number he chose will be readily apparent.

e.g.: 5 * 111111 = 55555

10 * 111111 = 1111110

300 * 111111 = 33333300

and so on.

If your friend chooses a two-digit number, where the sum of the two digits is less than 10, the result will be of the format HSSSSSF. Again, in these cases the number he chose will be instantly apparent; simply discard the "stem" and combine H and F:

e.g.:

15 * 111111 = 1666665

42 * 111111 = 4666662

51 * 111111 = 5666661

and so on.

In all other cases the result has "variable" digit(s) before the final, and the method requires a little mental calculation:

  1. Subtract the "variable" V or VV from the SAME number of digits in the "stem".
  2. Subtract the difference obtained in step 1 from the entire "head" H or HH.
  3. Append the final digit to the difference obtained in step 2. This is the number your friend used.

Some examples:

89 * 111111 = 9888879.
Form: HSSSSVF
Head = 9
Stem = 8888
Variable = 7
Final = 9
step 1:
   8 - 7 = 1
step 2:
   9 - 1 = 8
step 3:
   8 append 9 = 89.

128 * 111111 = 14222208
Form: HHSSSSVF
Head = 14
Stem = 2222
Variable = 0
Final = 8
Step 1:
   2 - 0 = 2
Step 2:
   14 - 2 = 12
Step 3:
   12 append 8 = 128

755 * 111111 = 83888805
Form: HHSSSSVF
Head = 83
Stem = 8888
Variable = 0
Final = 5
Step 1:
   8 - 0 = 8
Step 2:
   83 - 8 = 75
Step 3:
   75 append 5 = 755

569 * 111111 = 63222159.
Form: HHSSSVVF
Head = 63
Stem = 222
Variable = 15
Final = 9
Step 1:
   22 - 15 = 7
Step 2:
   63 - 7 = 56
Step 3:
   56 append 9 = 569.

408 * 111111 = 45333288
Form: HHSSSVVF
Head = 45
Stem = 333
Variable = 28
Final = 8
Step 1:
   33 - 28 = 5
Step 2:
   45 - 5 = 40
Step 3:
   40 append 8 = 408

Copyright 1998 by Bill Price
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