# Number puzzle: 4 is to 1, as 12 is to 2, as 86 is to ___? [closed]

I came across this puzzle and can't figure it out.
4 is to 1, as 12 is to 2, as 86 is to ___?

I wish I could clarify the question but this was the way it was presented without further clues or constraints.

The given answer is 3, but no explanation was provided.

• The trivial answer is that 4 is the 1st number in the puzzle, 12 the 2nd, and 86 the 3rd. But many - including me - won't find this a satisfying solution. :)
– dr_
Commented Sep 18, 2015 at 11:10
• Number of syllabes: Four, Twel-ve, Eigh-ty-six. It's a lateral thinking puzzle, the relation between the numbers is found through non-mathematical means Commented Sep 22, 2015 at 4:49
• Twelve is as much a two-syllable word as four, in that they are both one-syllable words. Commented Sep 22, 2015 at 12:15

One sequence that would work is the following:

f(x) = x^(x+1) + x + 2

Which gives:

f(1) = 1^2 + 1 + 2 = 4
f(2) = 2^3 + 2 + 2 = 12
f(3) = 3^4 + 3 + 2 = 86

However, having to figure that out from just the first two seems.. at least farfetched.

• I agree, but your solution gives me peace of mind. Thanks! Commented Sep 18, 2015 at 11:23

I have an alternate way of looking at it. The derived numbers 1, 2, 3 are

the count of the unique digits in the prime factorization of the original numbers.

Here is my explanation:

4 is factorized as $2\cdot2$, and the number of unique digits is 1.
12 is factorized as $2 \cdot 2 \cdot 3$, and the number of unique digits is 2.
86 is factorized as $2 \cdot 43$, and the number of unique digits is 3.

There can be many different answers. Without additional clues or constraints we can't be sure what the questioner intended.

Examples:

Polynomials of many different degrees could fit.

Factorising leads to some possibilities. We could suggest 86 -> 2, as 86 has 2 prime factors.

You could ask how many pen strokes are needed. 4 needs 1, 12 needs 2, 86 needs 2. Hence 86 -> 2 for a different reason.

• 33x^2-91x+62 fits for this question Commented Sep 18, 2015 at 13:13