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The Magic Cafe Forum Index » » Magical equations » » Why does this work (0 Likes) Printer Friendly Version

drkptrs1975
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Elite user
North Eastern PA
452 Posts

Profile of drkptrs1975
Take any 3 digit number, reverse the digits, subtract smaller from the larger, then reverse the digits again, then add them up. You will always get 1089.

Here is a few examples.

741

Reverse is 147

take smaller from the larger
741-147=594

Reverse them agian, this time add them

594 + 495

that will give you 1089


another one
758

857

857-758 = 99

099

990

099+990=1089
It works with all numbers, I cannot find a proof, nor a counter example. Why does this work.
stanalger
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Special user
St. Louis, MO
996 Posts

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Reverse the 3-digit number and subtract the smaller from the larger.
(100*a + 10*b + c) - (100*c + 10*b + a)
= 100*(a - c) + (c - a)

Note that after the first step, the value of the
central digit in the original number has already
become irrelevant.


Before we go on to step 2, let's rewrite:
100*(a - c) + (c - a)
=100*(a - c) - 100 + 100 + (c - a)
=100*(a - c - 1) + 100 + (c - a)
=100*(a - c - 1) + 90 + 10 + c - a
=100*(a - c - 1) + 10*9 + (10 + c - a)


Step 2: Reverse and add.

100*(a - c - 1) + 10*9 + (10 + c - a)
+ 100*(10 + c - a) + 10*9 + (a - c -1)
=100*(-1 + 10) + 10*(18) + (10 - 1)
=100*(9) + 180 + 9
=900 + 180 + 9
=1089
Scott Cram
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Inner circle
2677 Posts

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Another little-used aspect of this trick concerns the number you get after the first subtraction. Whatever number you get, the digits will always add up to 18.

About the only time I've seen this aspect used is by a book test created by Jim Steinmeyer.
graemesd
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Also I guess we all know that after the first calculation the middle number is aways 9 and the 2 outer numbers always add upto 9 eg 495, 198
see Banachek 'psychoogical subleties' for presentation ideas
idris
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St. Louis, MO
38 Posts

Profile of idris
Minor correction, the three digits must be different, at least the first and last digit. Usually the spectator is request to chose a number with three different digits. Obviously, if the first and last digits are the same the result of the first subtraction is 0.

Jerry
Jerry
Parson Smith
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1939 Posts

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This is an example of a root number of 9.
A couple of interesting things about nine.
Any # multiplied by 9 gives you a root of 9.
Any root of 9 multiplied by ANY whole # results in a root of 9.
Any string of #s added together and the sum subtracted from the original gives a root of nine.
There are many, many things that can be done with this.
Enjoy.
Peace,
Parson
Here kitty, kitty,kitty. Smile
+++a posse ad esse+++
Jaz
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NJ, U.S.
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Just a thought

Have someone do the calculation for the 1089 result.
Have them add 6645 and turn the paper upside down.
It looks like the word hELL.

Maybe even force two 6s a 4 and a 5 card so the 6654 number seems random.
Add some patter and who knows where it can take you. Hades likely. Smile
Mike Powers
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Midwest
2897 Posts

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If you are familiar with MOD arithmetic, here's a way to look at it. Since 10 MOD 9 = 1, all powers of 10 MOD 9 also equal 1. Thus all place values i.e. 1, 10, 100, 1000 etc = 1 MOD 9. Thus any number is equivalent (MOD 9) to the sum of its digits. That's why all multilples of 9 have the property that the sum of their digits is a multiple of 9.

(X MOD Y means the remainder when X is divided by Y. Thus 15 MOD 7 = 1 since 15 divided by 7 = 2 with a remainder of 1.)

Mike
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