Frogs or Toads?

First, start with Chris and Leroy. If both of them are frogs, they would both lie and their statements would be:

Chris: Leroy is a toad

Leroy: Chris is a toad

These statements would be the same even if they are both toads, only they will be telling the truth.

Both of them saying the other is a frog can happen only when one is a frog and the other is a toad. You can't say which is which though.

Conclusion: Either Chris or Leroy is a toad.

Now assume Brian is a toad. This would have two implications:

1. Since there is already one toad (either Chris or Leroy), the number of toads will become two.

2. Since Brian is a toad, Mike is a frog.

These two are inconsistent, considering that in this case, Mike's statement (that there are at least two toads) becomes true, while he comes out to be a frog. So, this case has to be rejected.

Change the assumption and consider Brian is a frog. This makes his statement wrong - Mike is also a frog! This makes Mike's statement wrong. Perfect! The world is in complete logical equilibrium.

That leaves us, unfortunately, with only one toad. Hence, 3 frogs.

*according to brian mike has to be frog

*From Chris and leroy one of them is frog and the other is toad

*as mike is frog so his statement isn't true

So the number of frogs is 3

Start with Brian. If he is a toad, he tells the truth, hence Mike is a frog. If Brian is a frog, he lies, hence Mike is a frog, too. Thus Mike must be a frog.

As Mike is a frog, his statement is false, hence there is at most one toad.

As there is at most one toad, at least one of Chris and LeRoy is a frog. But then the other one tells the truth, and therefore is a toad. Hence we must have one toad and three frogs. THus, answer is 3.

Let's start with Chris and LeRoy.

• Chris: LeRoy is a frog.
• LeRoy: Chris is a frog.

They can't be frogs, because then they would both be telling the truth. They can't be toads, because then they would both be lying. This means that one of them is a frog and the other one is a toad. (One is lying and the other one is telling the truth.)

What about Brian and Mike?

• Brian: Mike and I are different species.
• Mike: Of the four of us, at least two are toads.

If Brian is a toad, he is telling the truth. Then Mike turns out to be a frog that isn't lying. (Because there will be more than two toads among the four amphibians.) That's wrong!

If Brian is a frog, he is lying. Then Mike will be a frog too, and he will be telling a lie, just as a frog is supposed to do.

That leaves us with three frogs.

I went about this question by using binary numbers to represent each amphibian, 0 for a frog, 1 for a toad. So the combinations can be from 0000 -- 1111 where each combination represents an assumption of an amphibian being a frog/toad

e.g. 1000 means Brain is a toad, Chris is a frog, LeRoy is a frog and Mike is a frog. Now:

Chris and LeRoy both can't be of the same species because if they are both toads, then their statements are inconsistent and they can't be both frogs because then either one of their statemtns would have to be true in that case. So all combinations with X11X or X00X can be eliminated. That leaves us with 8 combos to be examined (down from 16).

Now come to Mike's statement. If we assume him to be a frog then all combos like 1100 etc (i.e where last digit is 0 and there are only 2 1's in the combo) can be ruled out because then in that case his statement would turn out to be true and he couldn't be a frog then. That gets rid of two more combos [1100, 1010], so we are down to evaluate 6

So first combo to be tried is

1. 0010 i.e LeRoy is a toad. So his statement that Chris is a frog holds. Chris being a frog gives the false statement that LeRoy is a frog since LeRoy is a toad. Brian's statement is false because Mike and he are same species so he can be a frog. Mike also being a frog, gives a false statement since there is only 1 toad.

We have found a solution but we can continue further like this and similarly other solutions are:

1. 0011 Can be rejected because then Brian's statement would hod true but we have assumed him to be a frog.

2. 0100 i.e Chris is a toad instead of LeRoy (as per prior explanation). A possible solution

3. 0101:Chris is a toad and Mike is a toad.Not possible because then Brian's statement would become true but we have assumed him to be a frog.

4. 1011: Can't be true because assuming Brain is a toad then his statement is inconsistent since MIke too is a toad in this assumption.

5. 1101: Same reason as above.

So the answer is 1 toad, 3 frogs and further either Chris/LeRoy is the lone toad.

You can complete this problem by basic logic operations, but it is a longer work. It is more easy to complete by induction.

Let me give a name to each statement.

Brian: "Mike and I are different species." .... (I)

Chris: "LeRoy is a frog." .... (II)

LeRoy: "Chris is a frog." .... (III)

Mike: "Of the four of us, at least two are toads." ... (IV)

Read each statement, and find the relations between them. And note that:

If II is true, then III should be false and If III is true then II is false. That expression is consistent because doesn't matter which of they is a frog or a toad. If Chris or Leroy is a toad then the another is a frog. (With II and III we have one frog)

If I is true then IV is false, but note that in this case we have 2 frog and 2 toads, that is a contradiction because the IV statement says "at least two of us are toads". This case is inconsistent. If I is false and IV is true there is a contradiction. Because if "Mike and Brian are different species" is a lie, then they should be the same species.

The last possibility is I and IV are both false. That make no contradictions. Thus there are 1 toad (from II or III) and 3 frogs (from I, II, III, IV)

Note that if a statement is false, the negation should be true. And if a statement is false the one who spoke it should be a frog.

If anyone enjoyed this problem, I'd recommend "What is the name of this book?" by Raymond Smullyan, which has a large number of increasingly devious problems along these lines.

There are three frogs. Either LeRoy or Chris is a toad; Mike and Brian are frogs

If Brian is a toad, that would mean his statement, "Mike and I are different species," is true.

If it is true, it would mean Mike is a frog (different species than Mike).

Since Mike is a frog, it means his statement, "Of the four of us, at least two are toads," is false.

~[[[REMEMBER: the phrase, "at least two," means "two or more."]]]~

Since Mike is a lying frog, it means there is only one toad, or no toad (0 toad, 4 frogs or 1 toad, 3 frogs).

Since Brian is already a toad, there cannot be 0 toads and 4 frogs. So this makes Chris and LeRoy both frogs.

Since Chris and LeRoy are frogs, their statements must both be lies. BUT if they are frogs, why are their statements truthful?... Therefore, this cannot work. ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

Therefore, Brian is a frog. Since Brian is a frog, his statement is a lie. Therefore, Mike is also a frog (same species as supposed to different).

Since Mike is a frog, his statement, "Of the four of us, at least two are toads," is a lie (remember "ate least two" means "more than two"). Therefore there's only one toad or 0 toad.

Chris and LeRoy cannot be both frogs because frogs do not tell the truth. Both of them are saying " _ is a frog". That means, one of them is lying, and one of them is telling the truth.

Therefore, Either Chris or Leroy is a toad:

If Chris is a toad, his statement, "LeRoy is a frog" must be true that LeRoy is a frog. Since LeRoy is a frog, his statement, "Chris is a frog" is a lie, which makes sense, because Chris is is toad (and vice-versa if LeRoy is a toad)

First, assume that Brain is a toad. Then, Mike's statement infers that Mike is a frog. So Mike's statement is false, i.e. only one is toad.

Now, take Brain a frog. Then also Mike's statement infers that Mike is a frog. So Mike's statement is false, i.e. only one is toad.

Chris and LeRoy are contradicting each other, so there must be one of each between the two.
Brian is denying to be the same species as Mike, so no matter the truth value of his statement, Mike will always be a frog.
Now, knowing that Mike is lying, then the claim he made about "at least there are two toads among them" is definitely false. It could be zero or one, but we already know either one of Chris or LeRoy is a toad, so the rest of them are frogs for a total of 4 - 1 = 3 frogs.

Chris and Leroy statements contradicts each other

that means one of them is Toad and the other is frog

Now If Brain's statement is true that will make him a Toad and makes Mike a frog therefor his statement must be false but that's not the case because we ended up having 2 Toads as Mike's statements says (Contradiction)

That means Brain is frog therefor his statement is false which means Mike also a frog

So there are $\boxed3$ frogs