TRIVIA / Puzzle # 1

I think you've asked the question wrong. Suppose that you have a very small cup. Then pouring blue color into red and then pouring mixed color back to blue bucket would only insignificantly change the colors in the buckets. So if question is really which bucket has highest ratio between RED and BLUE, answer is - red bucket off course.

I think that correct question is: Which bucket will have higher relative content of its original color (blue:red for blue bucket, red:blue for red bucket).

Answer to this modified (and harder) question can be determined in a funny way using so-called analysis of extremes.

First extreme - cup equal in size to bucket. First pour all blue into red, then pour half of that (one bucket) back. Result is that both buckets contain the same mix, so answer is: equal, 50% each.

Second extreme - cup size equals zero. First pour nothing from blue into red bucket. Then pour nothing back. Result is that both buckets contain only own original color, so relative content of original color is 100%.

Since problem looks linear in size of the cup, we can just interpolate these extremes with a straight line, which results in a global solution - for any cup in size less or equal to the size of the bucket, both buckets will contain equal relative amount of their original paint.

I have learned this kind of thinking from a university professor who passed away two months ago, so let this be an in memoriam post.

Don't think that this is not mathematical proof - it is. It is only less arithmetic, but none the less worth! Saves you from writing formulas explaining something that is obvious if you think deeper about the system you're solving.

Assuming:

I find the ratio of red:blue paint in the bucket originally full of red paint

is equal to the ratio of blue:red paint in bucket originally full of blue paint.

Mathematically the ratios are as follows,

Red Bucket - red:blue - b²/(b+c) : cb/(b+c)

Blue Bucket - blue:red - b²/(b+c) : cb/(b+c)

where,

b = volume of bucket; c = volume of cup

The workings rather long so i won't post it unless desired.

 

Sam

I suspect your friend is correct

My friend answered it differently. he answered that they will both be equal.

Let r = red;
Let b = blue;
Let 10 = # of cups in a bucket. //sample quantity


10r ~ 10b

// pour 1 cup from blue to red
10r + b ~ 10b -b
10r + b ~ 9b // simplifying

// pour 1 cup from red to blue
(10r + b) - ((10r + b)/11) ~ 9b + ((10r+b)/11)
10r + b - (10r/11) - (b/11) ~ 9b + (10r/11) + (b/11)

That's his solution

// to prove equality, we multiply both sides by 11.
11 * (10r + b - (10r/11) - (b/11)) ~ 11 * (9b + (10r/11) + (b/11))
110r + 11b - 10r - b ~ 99b + 10r + b
100r + 10b ~ 100b + 10r

Please comment.

Assuming Perfect mixing and equal sized buckets:

I think the red bucket has the highest ratio.  Here is the formula I used:

let's start with

Red Bucket = xr   Blue Bucket = xb

1:   blue --> red

xr + b   in the red bucket    xb - b  in the blue bucket

2:  red   -->  blue

red bucket:  (xr + b) - (xr + b)/y  =>  ratio of red to blue is always xr/b

blue bucket:  (xb - b) + (xr + b)/y  =   (x-1)b + (xr + b)/y => ratio always has a little more blue than xr/b

The ratio of red to blue in the red bucket is higher.

**one exception: if both buckets only have one cup of paint, then they are both equal ratio at the end.