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[Note: if you came to this post by way of DailyMail.co.uk, you should note that this article is based on a slightly different problem than the one they're talking about (look at the operations in the last line). Nevertheless, their proposed solution is still wrong (both for their problem and this one). Also, be sure to read our blog post about "Hard vs. Unsolvable!" Finally, if you're thinking about writing to us about this problem, please read the update at the bottom of this article.]

I should have known that after our write-up of the Apples, Bananas, and Coconuts problem went viral, we'd get asked to evaluate similar problems. Here's the latest one, and I wouldn't bother writing it up, except it has a fundamental difference from the banana problem. And people are taking the wrong lessons from the banana problem, and are still getting it wrong.

The correct answer to this question is, "No, I cannot solve this." There really is no other answer. If you came up with a numerical value, you were wrong (sorry!). Let me explain.

People who saw our write-up of the banana problem were quick to note that this problem has two similarities to that one: instead of two yellow flowers (as there are in the third line), there is only one in the last line, and instead of five petals (which it has in the second and third lines) the blue flower has only four petals in the last line.

The obvious (and erroneous) lesson to take here is that we're going to count petals instead of flowers. This leads people to the result of 25. So why is this wrong? Because the equations are not petal images; they are images of entire flowers.  We can calculate the value of a red flower, we can calculate the value of a yellow flower, and we can calculate the value of a blue flower with five petals, but we do not have any information with which to calculate the value of a blue flower with four petals. Unless we make the assumption* that the entire value of a flower lies in its petals. Ask a florist if he would sell a flower for the same price if the leaves and stem had been removed. I think you know what he'll say. 

Or we could make the assumption* that a flower is valued at the sum of its petals, stem, and leaves and also assume* that all petals, stems and leaves have the same value from flower to flower. In which case we could set up a system of three equations in 3 unknowns, with P representing the value of a petal, S representing the value of a stem, and L representing the value of a leaf:

36P + 3S + 3L = 60
22P + 3S + 3L  = 30
21P - 1S  - 3L = 3

And we need to find the value of 24P + 3S + 4L

If we subtract the second equation from the first, we get 14P = 30, or P = 15/7 (yuck).

You know what? I'm not going to follow through with this any further, because it's not a pretty system of equations, plus it's not inherently obvious that we should interpret the problem this way. So I think it's just a waste of my time.

* You will note that methods of attempting this problem will require us to make assumptions about the value of a stem, and the value of a leaf. And, furthermore, it requires us to make the assumption that each flower is worth the sum of its parts. Thus, the problem is unsolvable.

So I'm going to stick with my original answer: unlike the banana problem, this one is unsolvable.

So what is the big difference between the two problems? The difference is that in the bananas problem, each image was a representation of a number of fruits, so removing a fruit simply converts the image to another known quantity without fundamentally changing the nature of the objects represented, while in this problem the images are representations of entire flowers, so removing a petal doesn't convert it into a new known quantity; it fundamentally changes the nature of the object into something we don't know the value of.

That probably wasn't the answer the problem creator was looking for, but I don't really care - that's my answer, and I'm gonna stick to it!

Read more: Hard vs. Unsolvable!

Update June 17, 2016

This morning I received the following interesting email from Allan in Newcastle, Australia:

POLLEN PANDEMONIUM Why has nobody seen that the two yellow flowers are a multiplication (there is no + sign as there is in line 1), so that means one yellow flower is worth 1.4142.... (the sq.root of 2). The answer must therefore include .4142. I know you reject the following but . . .as the number of petals only matters for a blue flower, the answer is 25.4142.

Hi Allan! This is a fun way of looking at the problem. I don't think it's a correct way of looking at it, but it's still fun. I'll explain first why I don't think it's the correct way to look at it, but then, if you don't mind, I'll play a little game of "What if this WAS the correct way to look at it?"

Representing mathematical problems in a pictorial format is not a completely out-of-the-blue idea that the person who created this problem came up with. Math text books (especially at the younger grade levels) often use a pictorial format to illustrate math concepts. For example, in a lesson on place value, the number 27 might be represented by a picture of two rods (each representing 10), plus 7 squares (each representing one). In this case, the two rods represent exactly what they look like: two tens, not ten squared.

Similarly, if a textbook showed a picture of five dimes, a plus sign, and three more dimes, you should interpret that as a total of 8 dimes, not as 105 + 103.

So, even though I've never seen a rigorous definition of how we do pictorial mathematics, there's at least a "standard," and that standard suggests that a picture of two flowers really does represent two flowers, and not "flower squared."

Having said that, I've left a little loophole here; since I've never seen a rigorous definition, can we have some fun and see what happens if we choose to define it the way you did?

You said, in your e-mail, "the number of petals only matters for a blue flower," which is not strictly true. You see, one of the really important things we have to agree on in mathematics is that we be consistent in meaning, throughout a problem. Thus, if we need to deal at the "petal" level for blue flowers, in order to be consistent in notation, we need to deal at the "petal" level for all the flowers.

So the first equation is not 3r = 60, where r represents the value of a red flower. It's 3(12p) = 60, where p represents the value of a single red petal.

Except...if we agree to use your idea of using multiplication when no operation is presented, we now have to apply that at the petal level across the entire puzzle.

So it's not 3(12p) = 60; it's 3p12 = 60, because we have to multiply together all those petals!

So if 3p12 = 60,
p12 = 20

Which means p is actually the twelfth root of 20, which is about 1.284!

And then, of course, you need to follow the logic through on the other equations as well. :) I'm not going to do that; I'll leave it for readers to play around with the problem and see what answer they get. Thanks for your e-mail, Allan; this was a fun way to look at the problem.

Update June 21, 2016

When this post went viral, we started getting a large number of e-mails from helpful people who wanted to make sure we knew how to do the problem. This was exceedingly bizarre, since the article is pretty clear that it's not that we don't know how to solve the problem, but rather that there is not enough information to solve the problem without making unwarranted assumptions and therefore it cannot be solved. With the exception of the message from Allan in Newcastle (see June 17 update), each of these e-mails has done nothing but make one of the assumptions which was already addressed in this post. It makes it very evident that people are responding to this article without reading it. For this reason, we're no longer replying to emails about this post; Professor Puzzler is getting tired of simply redirecting people back to the article they should have read in the first place! Click here to read more about ambiguous math problems.

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