# How many spots do these frogs have?

You are approached by 5 spotted frogs who appear to be venomous, to avoid getting bitten by them, you must find out how many spots they have. The frogs with an even number of spots always tell the truth and the frogs with an odd number of spots always lie. It is known to you that all of the frogs have between 2 and 4 spots.

One frog says, "together, we have 17 spots."

The second frog says, "together, we have 16 spots."

The third frog also says, "together, we have 16 spots."

The fourth frog says, "together, we have 15 spots."

The fifth frog also says, "together, we have 15 spots."

How many spots do the frogs really have?

(Note: You are not able to count the spots on the frogs since they are moving too quickly.)

• They are five-spotted frogs. That is an odd number and they are all liars. 25. – MPS Mar 30 at 1:53
• @MPS For this reason, "It is known to you that all of the frogs have between 2 and 4 spots." – DenverCoder1 Mar 30 at 3:36
• @eyl327, is this information really necessary to solve? – Sigur Mar 30 at 12:19
• @Sigur It must be known that the liars all have 3 spots since it's possible that all of the frogs are lying. – DenverCoder1 Mar 30 at 12:23
• @eyl327, oh, I see. I was assuming (my fault) that both kind of frogs was there. Thanks. – Sigur Mar 30 at 12:24

Frogs

who make the same claim have the same parity of spot-count.

So

2 and 3 have the same spot-parity, and 4 and 5 have the same spot-parity.

So

these frogs between them have an even number of spots. So if the first frog is telling the truth then it has an odd number of spots too (17 minus an even number), but that's impossible because odd frogs lie.

So

the first frog is lying, and hence has an odd number of spots. So the total number of spots is odd, and therefore the second and third frogs are also lying.

Note that

the only possible odd spot-count is 3. So the first three frogs have 9 spots between them. If the other two are also lying then they contribute another 3 each for a total of 15 -- which is what they are saying! So that's impossible, so the other two are telling the truth and hence even-spotted but still contribute a total of 6 spots.

So:

Frogs 1-3 are all lying and therefore have 3 spots each. Frogs 4 and 5 are telling the truth; one has 2 spots and one has 4 spots. Unfortunately, I'm pretty sure you can't tell which is which from the information given.

• That's correct! Great reasoning! – DenverCoder1 Mar 29 at 11:10
• I believe that rot13(qrfcvgr lbh'er abg noyr gb pbhag gur rknpg ahzore bs fcbgf, ohg fgvyy pna qvfgvathvfu orgjrra "srj" (2) naq "znal" (4)). So you can calculate the number of spots for each frog. – trolley813 Mar 30 at 22:21

The frogs really have:

15 spots

Explanation:

Because different totals are given, we know there are at least three liars and at most two truth-tellers. If all were liars, there would be 15 spots, but this can't be as two frogs said there were 15. If only 1 frog was a truth-teller, there would be 4 liars (12 spots) + (2 or 4) = 14 or 16. But no frogs say 14, and two say 16, so this is not possible. There must be two truth-tellers.
The possibilities are:
2 2 9
2 4 9
4 4 9
, which add to 13, 15, and 17. Two frogs say 15, so this must be correct.

• Great job! This is exactly the logic I was expecting. – DenverCoder1 Mar 29 at 11:10

15 spots

Logic:

1 - All frogs cannot be lying or the answer would be 15, which two frogs state (and would therefore not be lying about)

2 - There can't only be 1 truth-teller or the answer would be even (4*3 plus an even number), and no unique answer is even

3 - There can be no more than 2 truth-tellers, since there are no more than 2 of any answer, therefore there are 2 truth-tellers

4 - The total therefore must be A) - an odd number (3 liars times 3 spots equals 9, plus two even numbers) and B) - given by two frogs (the truth tellers)

5 - Only 15 meets these two criteria

The frogs have:

15 spots

Because:

If all frogs were lying, they would have $$15$$ spots, but then they wouldn't have all been lying.

If frog $$1$$ was telling the truth, they would have either $$14$$ or $$16$$ spots, but then frog $$1$$ was lying!

If frogs $$2$$ and $$3$$ were telling the truth, then the frogs would have either $$13, 15$$ or $$17$$ spots, but then frogs $$2$$ and $$3$$ were lying!

Frogs $$4$$ and $$5$$ are therefore telling the truth - the possible number of spots are $$13$$, $$15$$ or $$17$$, and so it is $$15$$.

Starting

1 - All frogs have 2, 3, or 4 spots
2 - If a frog has 3 spots, he is lying, and if he is lying he must have 3 spots
3 - If a frog has 2 or 4 spots, he is telling the truth, and if he is telling the truth he must have 2 or 4 spots
4 - For any odd total, there must be an odd number of frogs with 3s (liars), otherwise the total would be even
5 - Frogs 2 and 3 are both either lying or telling the truth
6 - Frogs 4 and 5 are both either lying or telling the truth

Starting...

Let's assume that Frog #1 is telling the truth. That means he must have 2 or 4 spots. That means the remaining 4 frogs must have 13 or 15 spots. That means that one or three of them must have 3 spots. If all 4 or only 2 had 3 spots then the total would be even. This is impossible because 2 and 3 are either both odd or both even and 4 and 5 are either both odd or even, so this is impossible. Therefore we know frog #1 is lying and he has 3 spots.

We now have three options:

1 - The total is 16 (Frogs 2 & 3 are telling the truth and 4 & 5 are lying)
2 - The total is 15 (Frogs 4 & 5 are telling the truth and 2 & 3 are lying)
3 - All the frogs are lying

But...

Option 3 is not possible. If that were the case, all 5 frogs would have 3 spots giving a total of 15, that means Frogs 4 & 5 would be telling the truth. Therefore we know the total number of spots is 15 or 16, and two of the frogs are telling the truth while three are lying.

So...

Since we have an odd number of frogs lying and therefore having and odd number of spots, the total must be odd. Therefore Frogs 4 & 5 must be telling the truth, meaning Frogs 2 & 3 are lying and have 3 spots.

Finally...

Now we know that Frogs 1, 2, and 3 are lying and have 3 spots giving 9 total between them, and the total is 15 so the other two frogs have 6 spots between them. That means that Frogs
4 & 5 have 2 and 4 spots in some order. I don't see how it is possible to tell which one has 2 spots and which one has 4 spots with the information given.