# Thread: Three and Five to get Four

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## Three and Five to get Four

A chef has a recipe that requires exactly 4 liters of milk and offers you \$20 for it. You only have two similar cylindrical cans (one holds exactly 3 liters and the other holds exactly 5 liters). Your source of milk is metered and you are charged \$2 for every liter dispensed. Conservationist tax will charge you \$1 for each liter dispensed that not used in the recipe. With just the cans for measuring, what is the most profit you can make?

The cans are unmarked and start empty. Inaccuracies caused by minor things such as the thickness of the cans, loss of milk during multiple pourings, the ability to pour right to the rim, surface tension rise above the rim, etc. are negligible.

Note: There are multiple solutions that can get the 4 liters but I know of only one that turns a profit.
Last edited by ggchuck; 2019-Dec-09 at 10:51 PM. Reason: Add note and consistancy

2. Are you making the recipe in the 5-liter container?

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Not necessarily, though it could be. You start with empty cans so the milk would be the first ingredient.
(I see I called the containers both bowls and cans. I'm editing the post for consistency.)

4. Originally Posted by ggchuck
Not necessarily, though it could be. You start with empty cans so the milk would be the first ingredient.
(I see I called the containers both bowls and cans. I'm editing the post for consistency.)
Not necessarily? It makes a difference.

1) Dispense into the 3-liter container (\$6), pour it into the 5-liter container.
2) Dispense into the 3-liter container (\$6). Use it to fill the 5-liter container, leaving 1 liter in the 3-liter container. Pour that remaining 1 liter into your recipe, emptying the 3-liter container.
3) Pour from the 5-liter container into the 3-liter container, then add that 3 liters to your recipe (4 liters in the recipe).
4) Dispose of the remaining 2 liters in the 5-liter container (\$2 fee).

\$14 total spent, \$6 profit on the recipe.

If you don't have a separate container you're making the recipe in (eg, no place to put that 1 liter), then it's a much more difficult puzzle.

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Originally Posted by SeanF
\$14 total spent, \$6 profit on the recipe.
You got me on that one.
Using similar logic but a different order of pouring, I could get the 4 liters but I had a lot more waste.

Thanks to you, I now know of two solutions that turn a profit instead of just one.
There is still another way to get an even larger profit.

6. Originally Posted by ggchuck
There is still another way to get an even larger profit.
Multiple recipes?

Originally Posted by SeanF
1) Dispense into the 3-liter container (\$6), pour it into the 5-liter container.
2) Dispense into the 3-liter container (\$6). Use it to fill the 5-liter container, leaving 1 liter in the 3-liter container. Pour that remaining 1 liter into your recipe, emptying the 3-liter container.
3) Pour from the 5-liter container into the 3-liter container, then add that 3 liters to your recipe (4 liters in the recipe).
Then continue:

4) Pour the remaining 2 liters into the empty 3-liter container.
5) Dispense 5 liters (\$10) into the 5-liter container. Use it to fill the 3-liter container, leaving 4 liters in the 5-liter container. Use that 4 liters to make a second recipe.

Now, at this point, we've made two recipes we can sell for \$40, and we've spent \$22 on milk and have 3 liters to dispose of (\$25 total) for a total \$15 profit or an average of \$7.50 per recipe.

But, if we're doing multiple recipes...we currently have 3 liters in the 3-liter container, which is where we were on Step 1. So, we can just repeat the steps to make more recipes. We could repeat those steps ad infinitum - at any point we stop after step 3 and dispose of the remaining 2 liters.

We could approach the ideal maximum limit of \$12 profit per recipe. If we made 1000 recipes and sold them for \$20,000, we would have purchased 4002 liters of milk at \$8004 dollars for a total profit of \$11,996 or \$11.996 per recipe.

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Originally Posted by SeanF
Multiple recipes?
I would have to say no. Ingenious idea though.
If it was allowed, I could claim to know of an infinite number of solutions that are profitable.

If you wish, I can PM a clue or the solution.
You could then comment on whether the solution is logical or not and/or if the instructions are unfair/ambiguous without spoiling the puzzle for anyone else.
Last edited by ggchuck; 2019-Dec-10 at 03:48 PM.

8. Originally Posted by ggchuck
I would have to say no. Ingenious idea though.
If it was allowed, I could claim to know of an infinite number of solutions that are profitable.

If you wish, I can PM a clue or the solution.
You could then comment on whether the solution is logical or not and/or if the instructions are unfair/ambiguous without spoiling the puzzle for anyone else.

I only dispensed 6 liters total - you can't measure fractional liters, so the only way to get a better profit is to only dispense 5 liters. I don't see how you can measure 4 out of a dispensed 5.

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If the container in which you're mixing the recipe holds exactly 4 liters then you can dispense 5 liters, dump it into the recipe container, and let the 1 extra liter overflow. If the recipe container will hold more than 4 liters then this won't work.

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Originally Posted by Chuck
If the container in which you're mixing the recipe holds exactly 4 liters then you can dispense 5 liters, dump it into the recipe container, and let the 1 extra liter overflow. If the recipe container will hold more than 4 liters then this won't work.
Good idea, but no, I can't agree to let the recipe container be 4 liters.

I'll make you the same offer as I did to SeanF. If you wish, I can PM a clue or the solution.
You could then comment on whether the solution is logical or not and/or if the instructions are unfair/ambiguous without spoiling the puzzle for anyone else.

11. Order of Kilopi
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If the two containers are 'similar' in the geometrical sense then the 3 liter container could be put into the 5 liter container upside down so the 5 liter container would hold only 2 liters.

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Originally Posted by Chuck
If the two containers are 'similar' in the geometrical sense then the 3 liter container could be put into the 5 liter container upside down so the 5 liter container would hold only 2 liters.
You got it. The assumption that container thickness was negligible was also required.
- Nest the cans and hold firmly with the rims even with each other.
- Pour around the smaller can until the space available in the 5-liter can is full (2 liters).
- Pour the 2 liters into the smaller can and refill the available space in the 5-liter can the same way (2 liters).
- Go to the chef and collect your earnings. (Result: \$12 profit)
No need to add to recipe piecemeal.

13. Just a thought. What if you hold the containers at an angle, and fill each one until the surface of the milk goes from the bottom corner (a place where the bottom plate attaches to the side) to the rim? Then I think you will have 2.5 liters in one container and 1.5 in the other. And adding them gives 4 liters, without any waste at all. Given that the containers are cylinders, won't that work? And it's 8 dollars too.

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Originally Posted by Jens
Just a thought. What if you hold the containers at an angle, and fill each one until the surface of the milk goes from the bottom corner (a place where the bottom plate attaches to the side) to the rim?
Brilliant and elegant.
That eliminates the requirement that they have to be geometrically similar and the thickness of the cans wouldn't have to be negligible.

15. Originally Posted by ggchuck
Brilliant and elegant.
That eliminates the requirement that they have to be geometrically similar and the thickness of the cans wouldn't have to be negligible.
Yes, but it requires you to eyeball the fill level rather than using the fact that it spills over. That's only a shade better than just eyeballing 4/5 of the way up the larger cylinder.

For what it's worth, for your method, the two containers do not actually need to be geometrically similar, nor do they need to be cylindrical (or even symmetrical) - the smaller one just needs to fit entirely inside the larger one.

But saying they're "similar" is an easy way to ensure that the smaller one does fit entirely inside the larger one, without giving away that that's the real requirement.

16. Oh - If I wanted to nitpick anything, I'd argue that "Inaccuracies caused by minor things such as the thickness of the cans..." isn't really the same thing as declaring the cans to be negligibly thick.

That being said, that statement should've been a clue because it suggests an importance to the outside dimensions of the can...

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Originally Posted by SeanF
...it requires you to eyeball the fill level rather than using the fact that it spills over. That's only a shade better than just eyeballing 4/5 of the way up the larger cylinder.
It may be tricky, but it does give two distinct points for reference. The ability to eyeballing an unmarked container is suspect and negates the puzzle entirely.

Originally Posted by SeanF
But saying they're "similar" is an easy way to ensure that the smaller one does fit entirely inside the larger one, without giving away that that's the real requirement.
Exactly the reason the cylinders were chosen. It is important to specify a shape since something like similar hourglasses wouldn't work. To eliminate the solution Jens provided, tapered cups (similar truncated cones) could be specified instead of cylinders. Cups would also negate the inversion of the smaller container in Chuck's solution as the taper might preclude total immersion.

It was unfair to use the word "similar" in a conversational tone (as opposed to "geometrically similar"). In my defense, I did use it to describe a geometric shape.

Thanks to all who participated. I found aspects to the puzzle that I would never have considered otherwise.
Last edited by ggchuck; 2019-Dec-11 at 07:50 PM.

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