A Neptunian relays the following story: A Neptunian buys some gloves, and gets 28 grepleck change from his 100 grepleck note. He wears his new gloves, enjoying having all his fingers warm for a change (it’s cold on Neptune!), but tears one of the gloves. 
He recalculates that, since that glove must be thrown away, he has now paid 12 grepleck for each warm finger. How many fingers per hand do Neptunians have?
I don’t see how it can be answered uniquely:
He paid a total of 72 greplecks for the gloves, and at 12 greplecks per warm finger, that would mean that he now has 6 warm fingers left, and one hand worth of cold fingers. But we don’t know how many hands a Neptunian has.
But as I’m typing that, I think I see the trick:
[spoiler]The given numbers 28, 100, and 12 are not base 10, but in the Neptunian numbering system, which presumably has as a base the total number of fingers a Neptunian has. Now we’ve got a Diophantine equation on our hands (however many of those there are). To cut the problem down to size, I note that Neptunians have the digit 8, so must have at leas 9 fingers, and must have more than one hand, since he still has some warm fingers after losing one glove. Going through the possibilities:
9 fingers, 3 each on 3 hands: He gets 26DEC change from an 81DEC note, meaning that the gloves cost 55DEC. He has 6 warm fingers, and the money per warm finger isn’t an integer.
10 fingers: Already ruled out, since when I assumed base 10, I got 6 warm fingers
12 fingers: 32DEC change from a 144DEC note, meaning the gloves cost 112DEC, and he paid 14DEC for each warm finger. That means 8 warm fingers, and 8 is 2/3 of 12, so he could have two hands of warm fingers and one cold hand. This works.
For completeness, I should rule out any larger answers, but I’m not sure what the logical upper bound is. So I’m going to say four fingers per hand.[/spoiler]
As I get it , this boils down to the equation ( hands -1) x fingers = 6 .
(h-1) x f= 6
there are 3 possible combinations.
4 hand , 2 fingers
3 hand , 3 fingers
2 hand , 6 fingers
Formalizing the math a bit:
[spoiler]Let F be the number of fingers per hand, and H be the number of hands. Then the value of the note is (FH)^2, the value of the change is FH + 8, and so the cost of the gloves is (FH)^2 - FH - 8. There are F(H-1) warm fingers, and the price per warm finger is FH+2, so the equation becomes
(FH)^2 - FH - 8 = (FH-F)(FH+2)
Simplifying a bit, we have 3FH + 8 = F(FH+2)
Of course, we still have just one equation in two unknowns, so we still need to use Diophantine methods.[/spoiler]
And now I’m thinking that I did the algebra wrong, and my equation should be 4FH + 8 = F(FH+2), which would bean that we can’t find H at all, but we can find F = 4, as requested, even without explicitly requiring that it be an integer.
You’re assuming that number of fingers per hand is a constant.
Obviously, since otherwise the question would be meaningless.
14 k of g in a Neptunian glove?
Working from chronos’s assumption that the counting base is FH.
I come up with Neptunians have 24 fingers across 6 hands, so 4 fingers per hand.
All in base 10:
Bill denom = 576
Cost of gloves = 520
Change = 56
Cost/Warm finger = 26
Number of warm fingers = 20 (520/26).
(h-1)f=20
HF=24
HF - f = 20
24-f = 20
f=4
(h-1)4=20
h=6
6 hands, 4 fingers each.
Chronos got it. There isn’t enough information in the problem to tell you the number of hands, and there isn’t enough information in the problem to tell you
the base of the numbering system even though at first blush it looks like the problem is waaaay overspecified.
I first heard this problem in 1993, and have been searching for the correct wording since 2000. What I like about this problem is that the numbering system itself gives information about the number of fingers. That’s just such an unexpected source for that information!
I don’t get it. Is the answer “6 fingers per hand” wrong? If so, why?
Wouldn’t you need more info about the nature of the tear? As an earthbound humanoid I’ve torn gloves in such a manner that 0 through 5 of my fingers have remained warm.
Not really, since the puzzle specifies that he threw the torn glove away. So it is not warming any fingers.
Oh hell no.
Yes it is wrong, because base 10 counting provides several answers. The neptunian could have 7 hands and 1 finger per hand, etc etc.
I get it now. Similar to the problem of the three children, product of their ages is 36, sum of their ages is house number across the street, oldest has blue eyes - you need to find a way where there is no possibility of several solutions being right.
To be honest, my first thought on reading the puzzle was that it was kind of silly for the Neptunians to have a hundred unit bill, since why would they necessarily use base ten? It was when I realized that 100 might not be one hundred that I got it.