really hard puzzle, noone has solved yet...

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Since the XXX is derived from the 3 sets of given numbers and the ### is derived from these as well as the XXX, doesn't it follow that the ### is derived soley from the 3 sets of given numbers?

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not neccisarily.

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Then by "and the given numbers" you must mean more than the specific numbers that were used in the calculation of the corresponding XXX (i.e., some sort of progressive calculation in which numbers other than the corresponding numbers are used to calculate the ###). Otherwise, this doesn't seem to make any sense.

PS - Found some sort of calculation that worked for the 125 of RED125 and the 167 of CAT167, and was utterly heartbroken when it did not work for the 195 of ANT195.

I think I'll take a break.

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"74 76 68" are the only given numbers that are used to get the ###. that is all ill say for now.

what were the rules that worked for those 2 numbers? you might be on the right track.

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My rule, which as it turned out only worked for the first two number/letter combos, only works when you employ different rules for odd and even sums of the three preceding numbers.

Let x = even number
Let y = odd number

When sum(a,b,c) = x (where the sequence is "a b c LLL###"), then ### is calculated as follows:

### = 2x - (x/2 +1)

When sum (a,b,c) = y (where the sequence is "a b c LLL###"), then ### is calculated as follows:

### = 2y - ((y-1)/2)

Examples:

37 24 23 RED125
sum(37,24,23) = 84
84*2 - (84/2 + 1) = 125

30 49 32 CAT167
sum(30,49,32) = 111
111*2 - ((111-1)/2) = 167

So I got very excited until I was faced with 51 38 57 ANT195, which yielded the following result:

sum(51,38,57) = 146
146*2 - (146/2 + 1) = 218

The only thing I can safely say is that I'm fairly certain that the sum (or at least some linear combination of the preceding numbers) is involved in the formulation of ###. This must be the case because as the preceding numbers increase in magnitude as a group (i.e., the preceding numbers for RED < the preceding numbers for CAT < the preceding numbers for ANT, etc.) so do the ###s. This might be stating the obvious, but it rules out any other trivial use of these preceding numbers, such as summing all the digits within the numbers until you get a number less than 10, as this would not produce a direct correlation between the preceding numbers and the ###s.

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Was I close?