I just couldn't figure out how to solve this with truth tables and values. My reasoning went:
If "X is red" is true, then "Y is not red" must be false because only one statement can be true, so Y is red, so "X is red" is false after all. Since X is not red, X is either blue or white. If X is blue, then "Z is not blue" is true, so the only remaining colour Z can be is white.
Conclusion: X is blue, Y is red, Z is white. The first two statements in the original question are false, the third is true. Right?
If the above approach can be described in contrast to Boole's as "trial and improvement", it isn't too laboriously so I hope, since at least I didn't go through every possible combination of colours for the counters and truth values for the statements. I only tried out two statements "X is red" and then "X is blue" in the course of homing in. (Though admittedly it took a lot of cogitation before deciding on that course in the first place). I really look forward to an explanation of how the answer can be calculated using AND and OR Boolean style without this process of elimination.
Interesting that Boole mathematised logic, but later thinkers in the 19th and early 20th century tried to logicise maths.