many hours applying these transformation rules to strings. However, it might be quicker to find a property that is invariant to all rules (i.e. that isn't changed by any of them), and demonstrates that getting to MU is impossible. By looking at the puzzle from a logical standpoint, one might realize that the only way to get rid of any I's is to have three consecutive I's in the string. This makes the following invariant interesting to consider: The number of I's in the string is not a multiple of 3. This is an invariant to the problem, if for each of the transformation rules the following holds: if the invariant held before applying the rule, it will also hold after applying it. Looking at the net effect of applying the rules on the number of I's and U's, one can see this actually is the case for all rules: {| class=wikitable |- ! Rule !! #I's !! #U's !! Effect on invariant |- | style="text-align: center;" | 1 || style="text-align: right;" | +0 || style="text-align: right;" | +1 || Number of I's is unchanged. If the invariant held, it still does. |- | style="text-align: center;" | 2 || style="text-align: right;" | ×2 || style="text-align: right;" | ×2 || If n is not a multiple of 3, then 2×n isn't either. The invariant still holds. |- | style="text-align: center;" | 3 || style="text-align: right;" | −3 || style="text-align: right;" | +1 || If n is not a multiple of 3, n−3 isn't either. The invariant still holds. |- | style="text-align: center;" | 4 || style="text-align: right;" | +0 || style="text-align: right;" | −2 || Number of I's is unchanged. If the invariant held, it still does. |} The table above shows clearly that the invariant holds for each of the possible transformation rules, which means that whichever rule one picks, at whatever state, if the number of I's was not a multiple of three before applying the rule, then it won't be afterwards either. Given that there is a single I in the starting string MI, and one that is not a multiple of three, one can then conclude that it is impossible to go from MI to MU (as the number of I's will never be a multiple of three). Invariant set A subset S of the domain U of a mapping T: U → U is an invariant