How Many Flowers?
All of Sally's flowers, except two, are roses.
All of her flowers, except two, are tulips.
All of her flowers, except two, are daisies.
How many flowers does Sally own?
(there are 2 possible solutions)
All of her flowers, except two, are tulips.
All of her flowers, except two, are daisies.
How many flowers does Sally own?
(there are 2 possible solutions)
posted by Zaux at
5:40 AM
17 Comments:
3. I'll be back though.
Sally has either got 1 rose, 1 tulip and 1 daisy, or 2 of an unspecified flower or flowers.
Morning Chris ...
right you are ...
you watching the Super bowl today .. or do you not give a flip about American football? ... :)
All of Sally's flowers, except two, are roses. (A)
All of her flowers, except two, are tulips. (B)
All of her flowers, except two, are daisies. (C)
Let F be the total number of flowers Sally owns.
Let R be the number of roses Sally owns.
Let D be the number of daisies Sally owns.
Let T be the number of tulips Sally owns.
Let A be the total number of all other types of flowers that Sally owns. (The question does not state that Sally owns only roses, daisies and tulips.)
By definition:
F=R+T+D+A (i)
First case scenaro, assume Sally only owns roses, tulips and daisies, ie A=0.
From (i):
F=R+T+D (ii)
From (A), (B) and (C) respectively:
F=2+R (iii)
F=2+T (iv)
F=2+D (v)
(iii), (iv) and (v) give:
2+R = 2+T = 2+D
R = T = D (vi)
We can now interchange R, T and D in any equation we like, as we know they are all equal to each other.
Using this method, (ii) can be changed to:
F=3R (vii)
From (iii) and (vii) we get
3R=2+R
2R=2
R=1
Therefore from (vi), T=1 and D=1
Enter these values for R, T and D into (ii) gives
F=3
So the first solution is that Sally owns 3 flowers which are one rose, one tulip and one daisy.
Now consider the second scenario, where A is non-zero.
(iii), (iv), (v) and (vi) still hold true so (i) can be changed to:
F=3R+A (viii)
(iii) and (viii) give:
3R+A = 2+R
R+A=2
A=2-R (ix)
(iii), (iv) and (v) can be written as:
F-2=R
F-2=T
F-2=D
substitute for R,T and D in (i) gives:
F=(F-2) + (F-2) + (F-2) + A
F=3F-6+A
A=6-2F (x)
(iii) and (x) give:
A=6-2(R+2)
A=6-2R-4
A=2-2R (xi)
(ix) and (xi) give:
2-R = 2-2R
-R=-2R
R=0
(vi) tells us that T=0 and D=0
(xi) gives A=2
So the second solution is that Sally owns 2 flowers, neither of which are roses, tulips or daisies. Maybe she has 2 orchids.
hi Simon ...
nice analysis ...
Thanks! I had the answer before any posts were made but instead of just saying "2 or 3" (Chris...) I decided to provide a proof! ;)
Wow that was mouthful, Simon, but nice. I couldn't really figure out another answer. Only 2 is what i calculated yet.
SpArKo
This post has been removed by the author.
Reposting as this is slightly better than my last version.
Hi Simon. I nearly always publish my reasoning (your ;) tells me
that you knew that).
Let f = total number of flowers, r, t, d and a be the number of
each type. Let (1) f = r+t+d+a. In the sequence specified in the
question, we obtain:
(2) t+d+a = 2, (3) r+d+a = 2, (4) r+t+a = 2.
(2)-(3) => t-r = 0 => t = r,
(3)-(4) => d-t = 0 => d = t. So d = t = r.
Subst t for d in (2) => 2t + a = 0
So either a = 0 and t = 1 or a = 2 and t = 0.
If a = 0 then r = t = d = 1 and f = 3
If a = 2 then r = t = d = 0 and f = 2
Of course a could be replaced by x+y and if a = 2 could have
x = 2, y = 0 or x = 0, y = 2 or x = y = 1.
Hi Chris - only a bit of gentle ribbing! I think you have a mistake in your analysis though-
Where you say "Subst t for d in (2) => 2t+a=0
it should say 2t+a=2 maybe?
Your reasoning would then be correct in yielding the only non-negative integer solutions a=0,t=1 and a=2,t=0
My proof took a little more space as I decided to provide algebraically complete solutions. (I agree that 2t+a+2 is pretty simple to solve by trial and error with non-neg integer values but this is not strictly a proof!)
;D
Sorry I had a mistake too! Of course I meant "2t+a=2" both times!
3 because the other 2 are not the specified flower therefore there is one of each making 3 flowers total.
Posted by The Godson
Hi Simon. Well spotted; it is/was a typo.
I'm not sure that I understand the rest of your comment though. You made an initial assumption that a = 0 without any explanation as to why you felt that was OK to do. A priori, it may even have led to a contradiction.
I don't believe that it is possible to show that 2t + a = 2 leads to a = 0 and t = 1, or a = 2 and t = 0, without resorting to trial and error (or what, to me, is the same thing, a set of tables). You can say that t and a are integers, as you assume that the problem doesn't allow for fractional flowers. If it did, then the problem would have an infinite number of solutions.
sally only has one rose and one tulip
daisies are weeds total flowers 2
A weed is a plant or flower that grows where you don't want it to.
... and more importantly, Sally cannot only have 1 tulip and 1 daisy in the given problem. She'll have to have two other types of flowers.
I vote 3.
That's before I read what everyone said about it
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