A mathemagical trick
This is a trick with numbers that you may like to try on
others, so indulge me and try it for yourself; you may be amazed. I even
explain how it works at the end.
- Think
of a three digit number, with each digit different (you can write it down,
because I won’t peek) - Get
a second number by reversing the number you decided on in <i
style='mso-bidi-font-style:normal'>step 1</i>. (write that down too) - Calculate
the difference between these two numbers; that is subtract the smaller
number from the larger number. (you’re glad you got some paper to do the
working out on, now) - You
now have a third number. If it is less than 100 make it back up to a three
digit number by sticking enough zeroes on the front. - Reverse
the three digit number you had at the end of step 4. - Add
the numbers from steps 4 and 5
together.
Scroll down for the answer.
The result of all this arithmetic is 1089.
If you did not get this check your subtraction, addition and
that you followed the instructions properly.
For an explanation scroll further.
Step 1.
Choose three different digits to make a three digit number.
I shall choose a,
b and c, to make up the number abc,
Which represents the number 100a + 10b + c
Step 2.
Reverse the number in step 1.
So we now have the number cba.
Which represents the number 100c + 10b + a
Step 3.
Find the difference, by taking the smaller number from the
bigger one.
We’ll assume that a is greater than c, so that the first
number is bigger.
(if a is smaller
than c, we will go back to step 1 and switch the numbers we chose
for a and c, this will not change anything, because we still end up with the
same pair of three digit numbers, except now a is bigger than c)
The resulting difference we are looking for is abc – cba,
In full this is:-
(100a + 10b + c) – (100c + 10b + a . . . . . . which becomes
100a + 10b + c –100c – 10b – a . . . . . . which
evolves in to
99a – 99c . . . . . . . . . . . . . . . . . . .which
can be further simplified to
99 (a – c)
To simplify the next steps we will replace (a – c) with d
We know that d is
a digit in the range 1 to 9 inclusive, because a is greater than c and
the most a can be is 9 and the least
c can be is 0.
So now we have 99d.
We need to reconstruct this in the form of a three digit number before we
continue.
99d = 100d – d . . . . . . . . Ok, but we can’t have
negative digits, so we’ll
. . . . . . . . . . . . . . borrow one of
the hundreds to help.
99d = 100(d – 1) + 100 – d . . . . . . . . . so the first
digit is (d – 1), which is a digit 0
to 8
= 100(d – 1) + 90 + 10 – d . . . . . . . . .this breaks the borrowed hundred into
a 9 for
. . . . . . . . . . . . . . . . . . . . . . . . . . .the
tens digit and 10 – d for the units
digit
Step 4.
Put zeroes on the front to make the result a three digit
number
Already done, because
The subtraction results in a three digit number with
(d – 1) . . . . . . . . . . . for its first digit (this might be
zero)
9 . . . . . . . . . . . . . . . . for
its second digit and
(10 – d) . . . . . . . . . . . for its third digit.
The value of this number is
100(d – 1) + 9x10
+ (10 – d)
Step 5.
Reverse the result of the three digit result of the step 4.
This reversal results in a number with the digits:-
(10 – d) . . . . . . . . . . . for its first digit
9 . . . . . . . . . . . . . . . . for
its second digit and
(d – 1) . . . . . . . . . . . for its third digit.
The value of this number is
100(10 – d) + 9 x 10 + (d – 1)
Step 6.
Add the numbers from steps 4 and 5.
Number from step 4 + Number from step 5
100(d – 1) + 9x10
+ (10 – d) + 100(10 – d) + 9x10 + (d – 1)
Rearrange to get the hundreds, tens and units together
100(d – 1) +
100(10 – d) + 90 + 90 + (10 – d) + (d – 1)
100(d – 1 + 10 – d) + 180 + (10 – d + d – 1) . . . The ds in the brackets cancel out
100 x 9 + 180 + 9
900 + 180 + 9
1089 
Addition notes:
i. . It does not matter if the middle digit of the original
number is the same as one of the other digits, because it disappears in step 3
and thus has no bearing on the result, but it is very important that the first
and last digits are different, because if they are the same the subtraction
results in zero.
ii. . I have observed that this trick works in any base
higher than 2, but you have to change the end result (for base b) to 1,0,(b-2),
(b-1)
iii. . When I do this trick, I write 1089 on a piece of paper
which I fold and reveal as the answer, when the brain work is done.
iv. . Have fun.