# The Cucumber Paradox

Suppose you have 100 lbs of cucumbers and these cucumbers consist of 99% water. You decide to leave the cucumbers in the sun for a while until they consist of 98% water. You bring the cucumbers back in, and you think, "Now the cucumbers should weigh a little less than they were before, right?" But, you try as hard as you can, and you still can't figure out how much they weight. How much do the cucumbers weigh? Good Luck!

This is a straight mathematical/logical puzzle, but be warned: it isn't very straightforward. :)

EDIT: When I say a percentage like 99% I meant 99 percent of weight is water. Sorry about the ambiguous question :)

EDIT #2: Apparently, the fact that 100lbs came out of nowhere is confusing, so I'm creating a back-up story...you are delivered 100 lbs of cucumbers that you bought on Amazon...because you did not use Amazon Prime, they dried up (shipping took a very long time), and some water disappeared (described above). What is the weight of the cucumbers now? (Please do not flag for "trying to promote something" because I lowkey do not work for Amazon.) XD

• @Phylyp Spoiler alert. – Mordechai Feb 14 '18 at 5:49
• -1: This question requires an assumption about what you mean by "percent". – Señor O Feb 14 '18 at 7:42
• 99% V/V or W/W? meaning as weight or as volume? – Oray Feb 14 '18 at 7:57
• I'm not following the part about not being able to figure out how much they weigh. You weighed them originally by some means and determined that they weighed 100 lbs. Why can't you use the same means to determine how much they weight now? – Michael Feb 14 '18 at 15:57
• An extremely old and well-known puzzle. – jwg Feb 14 '18 at 16:02

I'm gonna say:

50 lbs

Explanation:

(Assuming the 99% water is by weight)

- The % of X in the cucumbers is calculated as Weight of X / Weight of the cucumbers.

- In the start there's 99% water and 1% solids in 100 lbs of cucumbers.

- Only the Weight of water will change in the process. The Weight of solids won't change after evaporation.

- Going from 99% water to 98% water means the % of solids doubled from 1% to 2%.

- Recall that % of solids = Weight of solids / Weight of cucumbers.

- Since the % of solids was doubled, and Weight of solids didn't change, that could only mean the Weight of cucumbers is halved.

As such, the remaining total weight is 100 / 2 = 50 lbs.

• Too fast again :) – NL628 Feb 14 '18 at 2:51
• You won't mind if I wait for more responses before accepting, do you? – NL628 Feb 14 '18 at 2:53
• No problem with that c: considering you included lateral-thinking i'd love to see what others come up with as well! – votbear Feb 14 '18 at 2:54
• Ah yes, the power of being featured on hot network questions. That was a nice surprise to wake up to xD – votbear Feb 15 '18 at 2:03
• In all seriousness this was a nice puzzle though. Not too hard, simple enough to be understood/attempted/solved/explained without needing any papers to take note of stuff, the numbers are deviously designed to be intuitively misleading, and the solution gives that 'gotcha, that's brilliant' feel. It's a fantastic light riddle to tell verbally! Definitely one i'll keep in mind in case i can use it during conversations :) – votbear Feb 15 '18 at 2:06

By basic algebra,

100 lb has 1% solids, i.e. 1 lb. The final state has the same amount of solids, plus the remaining water. We'll call the weight of the remaining water $x$. So $$\text{dried weight} = x + 1$$ This water $x$ is now 98% of the weight, i.e. $$\frac{98}{100} = \frac{x}{x+1}$$ $$98x + 98 = 100x$$ $$98 = 2x$$ $$x = 49$$ So $$\text{total weight} = x+1 = 50 \text{ lb.}$$

I hope that was not too roundabout.

By intuitive logic,

1 pound of solids was 1%, but is now 2% of the total mass.
So if 2% of something is one pound, how much is 100% of that something?
50 times 2% makes 100%, so it must be 50 x 1lb = 50lb.

• Having added some math formatting to your post, might I encourage you to include your units on those numbers throughout all the equations? =) – jpmc26 Feb 15 '18 at 7:41
• Good Job. This is the correct answer, (although sadly not the first). Amazing job with the LaTeX. Congratulations! – NL628 Feb 16 '18 at 15:14

Given:

- 100 lbs. initial cucumber weight, 99% is water
- Only water removed resulting in water becoming 98% of final cucumber weight

Therefore:

- Initial weight of 'not water' is 1% of initial cucumber weight (100%-99%)
- Weight of 'not water' is 1% of initial 100 lbs. = 1 lb.
- The 1 lb. weight of 'not water' is 2% of final cucumber weight (100%-98%)
- 1 lb is 2% of 50 lbs. (1 lb/.02), your final cucumber weight!

• Hey, welcome to puzzling.stackexchange.com! Have you noticed these spoiler blocks in other's answers? You should be doing them to. Click edit on one of the answer to see how spoilers are done if you are not sure how to add them! – Andrew Savinykh Feb 14 '18 at 22:02
• Thanks for the help, I was just trying to figure that out! – Mick Feb 14 '18 at 22:09
• Welcome to Puzzling! (Take the Tour!) How does your answer add to the identical ones already given? You should always look at existing answers before providing one of your own, to ensure you are not just adding a duplicate. – Rubio Feb 14 '18 at 22:17
• Thanks Rubio. My answer may provide a simpler solution. Also, my reasons for posting was not just to solve the problem but also to play with the puzzling.stackexchange.com formatting since I've not posted here before. Pardon the intrusion! – Mick Feb 14 '18 at 22:34

I mean, it is tagged lateral thinking...

You had 100 lbs of cucumber plants and watered them so that they were 99% water. Then you left them in the sun to grow, and the cucumbers grew. They are only 98% water because the mass of the plants is included.

So the weight?

I’ll calculate this part when I have time. How many cucumbers does 100 lbs of plants produce?

• It was also stated pretty clearly that it's a straight mathematical/logical puzzle, but labelled lateral-thinking only because it isn't very straightforward. Don't use it as an excuse for wild flights of fancy. (The lateral-thinking tag was inappropriate to begin with and is now removed.) – Rubio Feb 14 '18 at 22:25
• @Rubio got it, thanks, sorry about that. – Quintec Feb 14 '18 at 23:08
• I've gotta give it to you. Entirely my fault for giving the lateral-thinking tag. I looked up its definition and took it too seriously. +1 lol – NL628 Feb 15 '18 at 1:12

(100 lbs)(453.592 g/lbs) = 45359.2 g

Pound to gram conversion

(45359.2 g)/(.99 g/cm^3) = 4581.374 cm^3

Density of water @ 30 C (or 86 F) for scenario's sake, so a warm day

density = mass/volume

disregard sig figs btw: (4581.374 cm^3)(98%/99%) = 4535.572 cm^3

Cucumbers shrink when the water evaporates

(4535.572 cm^3)(.99 g/cm^3) = 44901.026 g

Conversion to lbs:

(44901.026 g)/(453.592 g/lbs) = 98.9899 lbs

• Welcome to Puzzling.SE! Could you edit your answer to include spoiler tags, so as not to spoil the solution for anyone who wants to try the puzzle for themselves? – F1Krazy Feb 14 '18 at 10:57
• @F1Krazy - not a problem for this answer, because even under the assumption that the percentages were by volume instead of weight, this calculation is wrong. It reduces the amount everything by (99/98)%, not just the water. – Paul Sinclair Feb 14 '18 at 17:38

101 lbs

Cucumbers are alive, and they were growing... gain one pound of mass to the same 99 of water and you get 99/101 = 98%

• @Glorfindel Only because Rubio removed that tag at about the same time this answer was posted. For approximately 95% of this puzzle's lifetime (as of right now), it was tagged [lateral-thinking]. – Trevor Powell Feb 14 '18 at 23:35
• @TrevorPowell Right, thanks. That isn't really apparent from the LQP review queue :) – Glorfindel Feb 15 '18 at 6:42

Using lateral thinking (and building on the @thecoder16's solution)

They still weigh 100lbs

Reasoning:

You watered them and they grew. Now they have 2lbs of solids. If they contain 98 lbs of water that will be 98lbs/(2+98)lbs = 98% water.

Although, actually we don't know how much solid remains. Of course, with reasoning like this, there is an infinite number of solutions because:

You can assume they grew to have N lbs of solids. Now, to calculate their weight you use the algebraic formula: x / x+n = 0.98 where n is the new weight of the solids and x + n is their new total weight.

And if we do assume that the solids don't change then:

x / x+1 solved for x is: 49, and x+1 = 50 lbs

• (See comment on @thecoder16's solution) – Rubio Feb 14 '18 at 22:26
• @Rubio, that doesn't change the fact that there are an infinite number of solutions if the weight of the solids are allowed to change. Nothing in the question said they weren't. This solution also covers the case where N = 1lb and X = 49lbs leading to the accepted answer of 50lbs. – Octopus Feb 14 '18 at 22:28
• It doesn't say they aren't sea cucumbers, nor paper cut-outs of cucumbers. It doesn't say they aren't the cucumber shaped projections into our dimension of some extra-dimensional creatures that can change their mass and density at will. It does say it's a straight math/logic puzzle. It's not reasonable to invent facts not in evidence that are required to make a "solution" work, and use the excuse that the problem definition didn't exclude that possibility; why folks are asserting that these are growing cucumber plants, rather than just cucumbers as the plain text says, is beyond me. – Rubio Feb 14 '18 at 22:36
• @Rubio they don't have to be growing they could also be rotting/shrinking. My solution still stands as being more complete because it does not assume anything about the weight of the solid part of the cucumber. A solution that makes an unstated assumption cannot be complete. I recognized the puzzle immediately, but that's irrelevant, it does not mean that the commonly regarded correct answer is the only one. In a puzzle that states that the solids do not change, it would be. – Octopus Feb 14 '18 at 22:40
• @Rubio, and besides, none of your silly assumptions change the answer at all. – Octopus Feb 14 '18 at 22:51

Sigh - It's NOT SO SIMPLE. Here's my correction ...

The dry weight of the cukes is 1 lb. If they are 98% water and 2% dry cuke, then geez 1 lb is 2% of 50lbs. THAT'S surprising!

• You were warned "it isn't very straightforward". – Peregrine Rook Feb 14 '18 at 22:19
• Akk! You're right - correction to my answer... – Scott Sancetta Feb 15 '18 at 2:29
• ... making it now redundant with multiple other answers. (Maybe just delete and hope for better luck next time) – Rubio Feb 15 '18 at 3:21

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