Venn Diagrams: Maximum & Minimum

At a glance 9 rules

1. 1Regions as lettersGive each of the 8 regions a letter. Every region is a whole number, 0 or more; write the target and each total in letters, and the bounds come from 'nothing below 0'.
2. 2Surplus$\text{surplus} = a + b + c - U = E_2 + 2E_3$$a, b, c = \text{set sizes},\ U = \text{union},\ E_k = \text{in exactly } k$
3. 3Most in all three$\text{all three} \le \min(a, b, c)$$\text{union fixed: all three} \le \tfrac{1}{2} \times \text{surplus}$
4. 4Fewest in all three$\text{all three} \ge a + b + c - 2N$$N = \text{group};\ k \text{ sets: all } k \ge \Sigma\,\text{sizes} - (k - 1)N$
5. 5Pair totals cappedIf no pair total can pass $k$, use $U = a + b + c - \Sigma\,\text{pairs} + \text{centre}$: the centre is at most $U - (a + b + c) + 3k$, and at most $k$.
6. 6How big the union can be$\max(a, b, c) \le U \le \min(a + b + c,\ N)$$\text{nest the sets} \Rightarrow \text{low};\ \text{keep them apart} \Rightarrow \text{high}$
7. 7Pay with the centre or pairsA centre person uses 2 units of surplus, a pair person 1. For the most people in exactly one set, pay with the centre; for the most in exactly two, pay with pairs.
8. 8Every region non-emptyIf every region must have someone, park 1 in each region you are not pushing (1, 2, 3, ... if they must all differ), then push the rest into the target.
9. 9Build the diagramA bound is the answer only when a full diagram reaches it: fill all 8 regions with whole numbers, 0 or more, that match every total.