Punitive arithmetic: a million combinations into a hundred: 123456 = 100.

Greetings, ladies and gentlemen!

And now – time for a mathematical brainteaser. So without further ado – here it is! ->

How can you turn six arbitrary digits into a hundred?

That’s it!

Now, off you go to try to solve it; then come back later for the answers below. No peeking!

If you’ve come back scratching your head – fair enough. So here’s a bit more meat to this mathematical bone:

Is it possible, with any six consecutive digits (except for a leading zero) – using addition, subtraction, multiplication, division, exponentiation, roots, factorials, and/or parentheses – to get exactly 100? You can “glue” digits together, but you can’t rearrange them.

All right, off you pop again with your thinking caps firmly on…

And now – answers!…

So, how do you get 100 from “123456”? There are plenty of ways; for instance:

123456:

( – 1 + 23 ) * V ( 4 ) + 56 = 100 // for anyone new here: V(n) is the square root of n, done for convenience.
1 * 2 * ( – 3 * V ( 4 ) + 56 ) = 100
– 1 * 2 * ( 3 + 4 ) + 5 ! – 6 = 100
( 1 + 2 ^ 3 – 4 ) * 5 ! / 6 = 100
( ( 1 – 2 + 3 + 4 ) ! – 5 ! ) / 6 = 100
( – 1 + 2 + 3 ) * ( 4 ! – 5 + 6 ) = 100

Can we (“we”, btw – is moi, and/or the mathematical boffins over at our Fan Club) pull off the same trick with “654321”? Easily! ->

654321:

65 + 4 + 32 – 1 = 100 -6 + 5 ! + 4 + 3 – 21 = 100
( 6 ! / 5 ! + 4 ) ^ ( 3 – 2 + 1 ) = 100
( 6 + 5 – 4 + 3 ) ^ 2 * 1 = 100
( 6 ! – 5 ! ) / ( 4 + 3 – 2 +1 ) = 100
( 6 – 5 + 4 ! ) * ( 3 + 2 – 1 ) = 100

So, in this formulation the puzzle is solved completely and fully. Any six-digit sequence that doesn’t start with a zero can be turned into exactly one hundred using basic arithmetic operations.

But we didn’t stop there – and we turned into 100 all the six-digit strings starting with a single zero. Then we went on to solve all the other combinations starting with two, three, and more zeros. The answer under this extended interpretation of the puzzle conditions is as follows:

We weren’t able to turn the following six-digit combinations into 100:

000000 000107 000001 000113 000002 000116 000003 000117 000006 000121 000007 000123 000012 000126 000013 000130 000016 000131 000017 000160 000021 000161 000023 000170 000026 000171 000030 000201 000031 000211 000060 000061 001160 000070 000071

= only 35 combinations in total. Excellent result, colleagues!

And now, for your further arithmetic pleasure, here are a few elegant and/or complex solutions:

Elegant:

066660: – 0 ! + 66 + 6 * 6 – 0 ! = 100
088777: 0 + V ( V ( 8 + 8 ) ) + 7 × ( 7 + 7 ) = 100
088876: 0 + 8 + 8 + 8 + 76 = 100
267227: ( 2 – 6 ) * ( 7 – 2 ) * ( 2 – 7 ) = 100
870787: 8 + 70 + 7 + 8 + 7 = 100

Complex:

050606: ( – 0 ! + 5 ) * ( 0 ! – 6 ) * ( 0 ! – 6 ) = 100
085080: 0 ! + 8 + ( 5 + 0 ! ) ! / 8 + 0 ! = 100
160280: V 16 * ( V ( 2 * 8 ) )! + 0 ! ) = 100
300067: ( 3 – 0 ! ) * ( 0 ! + ( 0 ! + 6 ) * 7 ) = 100
620007: 6 ^ 2 + ( 0 ! + 0 ! ) ^ ( – 0 ! + 7 ) = 100

Very complex:

000057: ( ( ( 0 ! + 0 ! + 0 ! ) ! ) ! ) / ( 0 ! / 5 + 7) = 100
000221: 0 ! + ( – 0 ! + ( 0 ! + 2 ) ! ) ! – 21 = 100
071161: ( ( – 0 ! + 7 – 1 ) ! ) * ( – 1 / 6 + 1 ) = 100
071267: ( – 0 ! – 7 – 12 + 6 ! ) / 7 = 100
767177: ( – 7 + 6 ! – 7 + 1 – 7 ) / 7 = 100

Obviously (at least, to those of you, dear readers, who’ve gotten this far and are still with me:), it’s physically impossible to go through the entire million options by hand. You have to somehow optimize the task, and filter out the obviously solvable cases. And thus the discussion began…

In the end, we decided to split each “abcdef” six-digit string into two three-digit chunks – “abc” and “def” – and immediately filter out those where “abc” ⇒ 10, and “def” ⇒ 2 or 10 (since 10 * 10 = 10 ^ 2 = 100).

And we get… success! Right away, 77.5% of all six-digit combinations got filtered out. Then we started adding extra filters and crunching what was left. First we solved the remaining six-digit strings starting with “9xx”, and then moved downward.

Along the way we came up with lots of arithmetic sorcery; for example:

68: 6 ! / 8 = 90
565: 5 ! / 6 * 5 = 100
2662: 2 ^ 6 + 6 ^ 2 = 100
108: V ( V ( 10 ^ 8 ) ) = 100
656: ( 6 ! – 5 ! ) / 6 = 100
5116: 5 ! * ( 1 – 1 / 6 ) = 100 // 000360: ( 0 + 0 ! – 0 ! / 3 ! ) * ( 6 – 0 ! ) ! = 100
6715: 6 ! / ( 7 + 1 / 5 ) = 100 // 610067: 6 ! / ( 1 * 0 ! / ( – 0 ! + 6 ) + 7 ) = 100
1210: ( 1 / 2 ) V ( 10 ) = 100 // fractional root

670001: ( 6 ! / ( V ( 7 ! + 0 ! ) + 0 ! ) ) ^ ( 0 ! + 1 ) = 100 // V ( 7 ! + 0 ! ) = 71
776771: ( 7 ! – 7 ! / 6 ) / ( 7 * ( 7 – 1 ) ) = 100 // 7 ! – 7 ! / 6 = 4200
776777: ( 7 ! – 7 ! / 6 ) / ( 7 * 7 – 7 ) = 100
000267: – 0 ! – 0 ! + ( – ( 0 ! + 2 ) ! + 6 ! ) / 7 = 100

Using these methods, we solved all combinations that don’t start with zero or that start with a single zero, and then moved on to the more complicated cases: two, three, and more zeros at the start of the six-digit string. At that point we had to introduce fractional roots, which in normal arithmetic are used very rarely, and so on.

And with this all this arithmetic alchemy we ended up with a really wonderful result: we failed to get 100 from only 35 six-digit strings. Hooray!