Here is a neat math trick I learned from Alex Bellos’ book “Here’s looking at Euclid”.

**The Trick**

- Start by picking any three digit number where the first and last digits differ by at least two. For example 421.
- The revers it’s digits … 124
- Subtract the smaller number from the bigger number 421 – 124 = 297
- Reverse this result and add the two numbers … 297 + 792 = 1089

Now try this with any other three digit number whose first and last digits differ by at least two … say 519

reverse of 519 is 915

difference is 915 – 519 = 396

reverse of 396 is 693 when we add these together we get …693 + 396 = …

1089! (again!)

**The Proof**

This “trick” can be proved using algebra, lets start by letting the three digit number we select be “abc” where “a” is the hundreds value, “b” is the tens value, and “c” is the ones value in other words …

abc = 100a + 10b + c

The reverse of abc would be cba …

cba = 100c + 10b + a

When we subtract them from each other …

abc – cba = (100a + 10b + c) – (100c + 10b + a)

= 100 a + 10b + c – 100c – 10b – a

= 99a – 99c

= 99(a -c)

Remember the trick tells us that the difference between the first and last digit must be at least 2? The values of a -c will only range from 2 to 9, this would limit the possible results of 99 (a – c) = xyz to be …

99(2) = 198

99(3) = 297

99(4) = 396

99(5) = 495

99(6) = 594

99(7) = 693

99(8) = 792

99(9) = 891

Notice two things from the results shown above …

- The middle number is always 9 (y = 9)
- The sum of the first and last digit is always . (x +z = 9)

Now back to the proof. Let 99(a – c) be a three digit number “xyz”

xyz = 100x + 10y + z

While its reverse would be …

zyx = 100z+ 10y+ x

The last part of the trick asks us to add xyz to zyx which gives us ..

(100x + 10y + z) + (100z + 10y + x)

= 100x + x + 10y + 10y + 100z + z

= 101x + 101z + 20y

= 101(x + z) + 20y

Now remember that the first and last digits x and y always add up to 9 therefore (x + y = 9) and the middle number y is also always 9. Therefore we have …

101 (x + z) + 20y = 101(9) + 20(9) = 1089!

And there we have it! A magic trick explained with algebra!