# Understanding a famous riddle

This is quite a popular riddle in interviews and in general:

In a bouquet of flowers, all but two are roses, all but two are tulips, and all but two are daisies. How many flowers are in the bouquet?

The answer is $3$. My question is how exactly this is $3$? I suspect that answer lies in correct understanding of "all but two are".Please help me to understand the answer.

-
Since svenkatr gives a mathematical way of finding the answer, here is a way to understand the language of the problem once you know the answer. I hope you don't mind working backwards, once you know the answer is three. If you have a bouquet consisting of one rose, one tulip, and one daisy, then all but two are roses, since the tulip and daisy are not roses, and all but these two flowers in the bouquet leaves only the rose, which is indeed a rose. The same goes for the tulip and the daisy situations. – yunone Jun 1 '11 at 15:16

Let $n$ be the total number of flowers. When the problem says that all but two of the flowers are of one kind, it means there are $n-2$ flowers of that kind. Therefore, $n-2$ of them are roses, $n-2$ of them are tulips and $n-2$ of them are daisies. Assuming that this exhausts the list of flowers, we can write $n-2+n-2+n-2 = n$ which gives $n=3$

-
This makes complete sense,thanks :-) – Quixotic Jun 1 '11 at 15:16
I would only add that we don't know that these are all the flowers, so you really have the inequality:$$n - 2 + n - 2 + n - 2 \leq n$$ and $n\ge 2$. In particular, it is possible for $n=2$, by having a daffodil and a sunflower. – Thomas Andrews Jun 1 '11 at 15:28
@Thomas: I was just about to mention this! As you say, the assumption that we have "only" tulips daffodil's and sunflowers is not needed. – Eric Naslund Jun 1 '11 at 15:31
Or a dahlia and a snapdragon. – TonyK Jun 1 '11 at 15:44
@Debanjan Nothing in the question says that the bunch of flowers consists only of tulips, roses, and daisies. You could have only two flowers in your bouquet, with no tulips, roses, or daisies. (I just happened to pick two flowers, a daffodil and sunflower.) – Thomas Andrews Jun 1 '11 at 15:46

All but two mean's you have got to subtract $2$ from the quantity. Let $m$ be the total number of flowers. Then you have $$m-2 + m-2 + m-2 =m$$ and then solve for $m$.

-
Hi @Chandru, why did you delete the question on $SL(2,Z)$ and $A_5$? I just wrote a complicated answer, constructing the homomorphism etc. ;-) and I couldn't post it. That sucks. – Luboš Motl Jun 1 '11 at 17:52
@lubos: I humbly request you to delete these comments. – user9413 Jun 1 '11 at 18:07