buckets of water
The problem is that we have 3 buckets of water.
Bucket 1 has a capacity of 3 gallons and is initially empty. Bucket 2 has a capacity of 5 gallons and is initially full of 50 degree water. Bucket 3 has a capacity of 6 gallons and is initially full of 90 degree water. Now the "nonreallife" conditions: 1) We can only move water between these three buckets. 2) We can purposely spill water, but once it's spilled we can't pick it back up. (so don't try this inside your home) 3) The water temperature will not change due to air temperature, change in pressure, etc. 4) When you mix water from buckets of different temperature, all the water becomes the average temperature before our next transfer. (no waiting for heat transfer  mixing 1 gallon of 50deg and 3 gallons of 90deg produces 4 gallons of 80deg) 5) We can only be precise when completely filling or emptying a bucket (i.e. pouring 3 gallons from the 6 gallon bucket into the empty 3 gallon bucket) The puzzle is what integer temperatures between 50 and 90 can be produced? This is not my puzzle, so I only have a partial answer. [spoiler] HA! You thought this would be my partial answer, didn't you? [/spoiler] Also, this is the first puzzle I've posted so if anything seems unclear, let me know. 
A small question about mixing:
Consider the following  The 5 gal bucket is full. What happens if we attempt to pour the contents of the 6 gal bucket into it? Obviously, we will spill 6 gal. But at what temperature? Do we assume that there is absolutely no mixing (The original 5 gal @ 50 degrees remains in the bucket and the 6 gal @ 90 degrees gets spilled) ? or this  Pour out 3 gal of 90 degree water. The remaining water has an average temperature of (5*50 + 3*90)/8 = 65 degrees. Unfortunately, none of the water is at that temperature. However, we can mix the water in the 3 buckets a (infinite) number of times and get it all to the same temperature. Is the procedure allowed? 
[QUOTE=Wacky]A small question about mixing:
The 5 gal bucket is full. What happens if we attempt to pour the contents of the 6 gal bucket into it? Obviously, we will spill 6 gal. But at what temperature? Do we assume that there is absolutely no mixing (The original 5 gal @ 50 degrees remains in the bucket and the 6 gal @ 90 degrees gets spilled) ? [/quote] I've assumed this to mean no mixing so we would have a 50deg bucket.(another nonreallife conditional) [quote=Wacky] Pour out 3 gal of 90 degree water. The remaining water has an average temperature of (5*50 + 3*90)/8 = 65 degrees. ... we can mix the water in the 3 buckets a (infinite) number of times ... Is the procedure allowed?[/QUOTE] Let's allow these and mark any temperature only arrived at in this manner with some sort of special notation. 
An interesting problem. Lets say I pour 3 gallons of 90 degree water into the 3 gallon bucket. I then pour 3 gallons of 50 degree water back into the 6 gallon bucket producing a temperature of 70. Now I have 3 temperatures to work with.
3 gallons (in 3 gallon bucket) at 90 2 gallons (in 5 gallon bucket) at 50 6 gallons (in 6 gallon bucket) at 70 It looks like a simple recursive from there. Fusion 
[QUOTE=Fusion_power]3 gallons (in 3 gallon bucket) at 90
2 gallons (in 5 gallon bucket) at 50 6 gallons (in 6 gallon bucket) at 70[/QUOTE] Pour the 3G of 90D from 3GB into 5GB, get 5G of 74D or Pour 3G of 70D from 6GB into 5GB, get 5G of 72D Total accounted for 50, 70, 72, 74, 90. 
[QUOTE=tom11784]5) We can only be precise when completely filling or emptying a bucket (i.e. pouring 3 gallons from the 6 gallon bucket into the empty 3 gallon bucket)[/QUOTE]
I'm a little unclear about this point. I assume this suggests that when transferring water from one bucket to another or to the floor, you can only transfer either the entire contents of the initial bucket or the remaining space in the destination bucket, assuming either is possible. Is this true? Bucket 1 has a capacity of 3 gallons and is initially empty. Bucket 2 has a capacity of 5 gallons and is initially full of 50 degree water. Bucket 3 has a capacity of 6 gallons and is initially full of 90 degree water. Meaning that the following is not allowed: Pour 1 gallon of water from Bucket 2 into Bucket 1. If this true, I believe I have the answer. I'm going to try to work with it a little more to see if I've exhausted every permutation. 
[QUOTE=Qbert]Meaning that the following is not allowed:
Pour 1 gallon of water from Bucket 2 into Bucket 1. [/QUOTE] That is correct. You have no way to measure the 1 gallon at that point. 
Right  the only initial actions that can be made are to:
A) Pour 3 gallons from bucket 2 into bucket 1 B) Pour 3 gallons from bucket 3 into bucket 1 
[QUOTE=tom11784]Right  the only initial actions that can be made are to:
A) Pour 3 gallons from bucket 2 into bucket 1 B) Pour 3 gallons from bucket 3 into bucket 1[/QUOTE] Sorry, you left out two other options: C) Pour 5 gallons from bucket 2 onto the floor D) Pour 6 gallons from bucket 3 onto the floor (But they don't lead to any interesting possibilities.) 
I found those trivial .... and thus of no interest to people here

I'm still trying to think of some options, but I've been able to account for the following temperatures.
[spoiler]50, 55, 56, 57, 58, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 77, 78, 79, 80, 81, 82, 83, 84, 85, 86, 90[/spoiler] I'm still not sure whether or not the integer temperatures would follow a difference to 50 and 90 pattern. If so, notice peculiarities would be [spoiler]54, 59[/spoiler] All of these temperatures are attainable in a discrete number of steps. I don't believe using an infinite number of steps would result in anything worthwhile. 
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