cages is to add up the cages using 'clock' arithmetic (correctly, Modular Arithmetic modulo 10), in which all digits other than the last in any number are ignored. When two numbers are added together, the last digit of the total is not affected by anything other than the last digits of the two original numbers. Adding together a number ending in 7 and a number ending in 8 always results in a number ending in 5, for example. So, for example, 17 + 18 = 35 becomes, in clock arithmetic, 7 + 8 = 5. The biggest number an 'innie' or 'outie' can hold is 9, so adding or subtracting that value will change the last digit of the total in a way that no other value would - allowing the 'innie' or 'outie' to be directly calculated. Clock arithmetic has the advantage that you are only ever dealing with single-digit sums, rather than sums like, say, 58+27 - and even if the concept is initially unfamiliar, it rapidly becomes trivial. Example: A set of cages form a complete nonet with an 'outie'. The cages have values 8, 10, 14, 7, 14. Using normal arithmetic, those add up to 53. A single nonet totals 45, so the 'outie' must contain an 8. Checking that, using clock arithmetic on those values in turn: 8+0=8; 8+4=2; 2+7=9; 9+4=3. So the clock total is 3, meaning that the actual total also ends in 3 (which we've seen that it does). Any odd number of houses (in this case, 1 nonet) always have an arithmetic total ending in 5 - so, the only 'outie' we could add to change that 5 to a 3 is, again, 8. Clock arithmetic has the additional bonus that, when the final digits of two cage totals add up to 10 (13 and 27, for example), the pair will make no difference to the overall clock total, and can simply be skipped. Clock arithmetic should at most be used with caution for houses with more than one 'innie' or 'outie', when more than one set of values may result in the same final number, but may still be useful as a quick arithmetic check. Consistent numbers within combinations Even though some cages can have multiple combinations of numbers available, there can often be one or more numbers that are consistent within all available solutions. For example: a 4 cell