Paint mixing

ceptimus

A painter has a gallon tin (eight pints) full of yellow paint and another gallon tin half full of blue paint. He wants to mix the paints together so he can paint a room green. He reckons he will need at least 10 pints of paint to cover the walls.

Unfortunately, he doesn't have a mixing vessel large enough to hold all the paint - the only other container he has, besides the tins, being a pint glass.

Is it possible for the painter to mix the paint correctly using just these containers? What is the quickest method he can use?

Gokul43201

Does he need to mix them in the ratio 1 : 1 (Y:B) or in the ratio that he has them (2:1) ?

Edit : Ignore that.

Moonbear

Quickest? Okay, you take the can of blue paint, quickly invert it over the can of yellow paint, hold them tightly together, shake well, and then quickly turn them both right side up again. Sure, some will wind up sloshing onto the floor while doing this, but he has 12 pints and only needs 10, so can be a little sloppy about it. :-D

Njorl

pour 1 pint of yellow into pint can

pour 4 pints from yellow to blue, mix

pour 5 pints from blue can to yellow can, mix

pour 5 pints from the yellow can to the blue can, mix

pour 24/19 pints from blue can to yellow can. Pour 1 pint of yellow into blue can. Mix both.

Done

Gokul43201

How do u do that ? You can only pour integral numbers of pints.

vikasj007

i am not sure, but does he have to mix the two colours in the ratio of 1:1.

if this is so, then, i dont think that their is any real solution to the problem, because, he needs 10 pints to paint the whole room, that means that he needs 5 pints of yellow paint and 5 pints of blue paint, but he has only half gallon of blue paint i.e. 4 pints.
so how can he mix them in equal poportions to get 10 pints.

Gokul43201

No, 1:1 is not required...but all the paint needs to be the same color. You can't have - for instance - 5 pints of 2:1 and 5 pints of 3:1.

Njorl

That makes it much harder.

Let's see...

You need to end up with 2 to 1 ratios in each container

The only amounts you can have in your containers are:
8,4,0
8,3,1
7,5,0
7,4,1
6,6,0
6,5,1
(and permutations)

You can transfer :

1,2 or 4 from 8 to 4
5 from 8 to 3
1 2 or 3 from 7 to 5
1 from 5 to 7
3 from 7 to 4
1 from 4 to 7
1 or 2 from 6 to 6
3 from 6 to 5
2 from 5 to 6
1 from anything to zero

Is there a clever way to do this, or is it just awful trial and error?

Njorl

Gokul43201

I was thinking more along the lines of saving a pint of yellow; then keep intermixing till the difference (in color) between the two pails can be negated by pouring the pint of yellow into one of them.

ceptimus

Nice try but the two cans are different diameters, so this won't work.

And the cans and glass are also unevenly tapered so you can't tip either on the diagonal to get it exactly half full either (someone always tries that dodge )

Gokul43201

Ceptimus,

I'd like to repeat Njorl's question : "Is there a clever way to do this, or is it just awful trial and error?"

If it is the latter, please tell us that it can be done in 3 or 4 steps...else the number of possibilities get too large.

Finally, do we really need the whole gallon of yellow ? When I was walking to luch yesterday, I "thought" I figured a way in my head, that involved throwing out 2 pints of yellow in the beginning. Can't seem to recreate it since, to know if it was really correct. Only recall having one pint that was either 2/3 or 4/5 blue and transfering this at the end...extremely annoying !

ceptimus

There are at least two relatively simple solutions that both only involve a small number of steps. There is no trick or wordplay involved. As long as you end up with at least 10 pints of uniformly mixed paint, you have solved the puzzle, so you can throw away up to two pints of paint if you wish.

The Bob

I think I have it.

Right this is what you do:
1. Pour 1 pint of blue paint into the glass and throw it away. (Yellow:8 pints - Blue :3 pints)
2. Then pour another pint of blue paint into the glass and throw it away. (Y:8 - B: 2)
3. Then pour one blue pint into the glass and keep it (Y:8 - B:1 - Glass:1 (Blue))
4. Then pour 4 pints of yellow paint into the blue tin (I assume you can see if it is half or not by using your eye) [Y:4 - B:5 - G:1]
5. Then pour the blue paint from the glass into the yellow tin and that is it. (Y:5 - B:5 - G:0) and the concentrations are 1 part blue to 4 parts yellow. There is also 10 pints of paint.

Hope it is right, if not I will have to try harder.

The Bob (2004 ©)

Gokul43201

No, you can not !

Njorl

You know, I completely missed that "10 pints" bit. I've been trying to get 12 pints.

Njorl

Gokul43201

Still doesn't change that fact that it is basically a trial-and-error problem.

Njorl

Discard 1 pint yellow with pint glass, repeat.

use pint glass to transfer 1 pint yellow into blue can (1y 4b)
pour one pint yellow into glass and hold in reserve.

We now have (4y) in one can (1y,4b) in another and (1y) in the glass

Now dump all you can from blue can to yellow can.
You get (4.8y,3.2b) (.2y,.8b) (1y)

Dump pint glass into blue can to get 60/40 ratios in both cans.

It was just work, for the most part.

I figured we either dump 2 yellow, or 2 blue. We are also going to be dumping back and forth winding up with 8 pints in the receiving can, so we need a 4.8 to 3.2 ratio or a 6.4 to 1.6 ratio. Either way we need fifths of pints. After that, I just plugged away.

Was there some clever way to get the answer?

Njorl