# Solve the Unconventional Equality

Can you solve the following equality?

$432 = 915$
$944 = 850$
$211 = 403$
$837 = 927$
$735 =\: ???$

$734 = 5100$
$736 = 736$
$835 = 741$

HINT 1:

There are some numbers that will produce an error if run through the algorithm. An example of a number that would cause an error is $571$. Another example is $435$.

Super mega bonus hint: I'll go ahead and run a few requested numbers through the algorithm so you can see what the results are! The first 3 or so requested numbers will be added to the OP for everyone's guessing pleasure. I will, however, frown and stick out my tongue at you if you ask for the result of $735$.

• If you take the max of the two sides, you get a decimal extension which likes to having twins as repeating numbers :P – Conor O'Brien May 20 '15 at 23:17
• Let's see, by the principle of explosion we can get the right answer. – Milo Brandt May 21 '15 at 0:01
• Can we get a hint or something? – JLee May 21 '15 at 20:34
• What about 735? – Ian MacDonald Jun 1 '15 at 18:23
• @IanMacDonald: >:P – Bailey M Jun 1 '15 at 18:32

642

The first digit (as rand al'thor said), is:

The sum of the digits of the left hand number, so 7+4+2 = 15, 1+5 = 6

The other digits:

The first digit of the left hand side is the number base of the other digits. Convert them into the base given by the first digit of the right hand side
For the first example: 4+3+2 = 9
324 = 159
432 = 915

So for 735:
357 = 426
571 produces an error because 7 doesn't exist in base 5.

Hopefully that all makes sense!

• Perfecto! I wasn't expecting this one to take as long as it did. Well done! :) – Bailey M Jun 2 '15 at 15:28
• Thanks! It was a great puzzle! Probably wouldn't have got it without the first digit already having been solved, though. Also the helpful clue that the first digit was the only one that could be solved independently – LogicianWithAHat Jun 2 '15 at 15:30

The first digit is

6

because

the first digit of the right-hand number is always the digit sum of the number on the left (4+3+2=9; 9+4+4=17 and 1+7=8; 2+1+1=4; 8+3+7=18 and 1+8=9; 7+3+5=15 and 1+5=6)

The second digit is:

1

because

the second digit of the right-hand number is always the sum of the second and third on the left minus the first on the left - unless this figure is negative, in which case (too little data to tell, but it could be) the first digit on the left minus the second (3+2-4=1; 9-4=5; 1+1-2=0; 3+7-8=2; 5+3-7=1)

[WIP]

• Another way to say it is that the first digit is the value mod 9. Perhaps the other digits are with other mods? – xnor May 20 '15 at 21:23
• @xnor You reckon Doorknob might have the other digits? ;-) More seriously, I thought of that, but 211 is prime, so it wouldn't work for the middle digit (unless the base is 211). – Rand al'Thor May 20 '15 at 21:24
• @xnor BTW, congrats on your extremely round figure in rep (and on reaching 5 figures)! – Rand al'Thor May 20 '15 at 21:50
• You've done a spectacular job identifying the first digit! Don't be led astray by the fact that the first digit was solved by itself, however. :) – Bailey M May 21 '15 at 20:38
• @BaileyM Does that imply that a number cannot be solved just using its own features? – Aggie Kidd May 28 '15 at 22:11