pair-sum-invariant
Partition a universe into forced complementary pairs (or up/down segment pairs) so that any legal configuration has a fixed total. Key move: re-pair the items so the constraint becomes a closure/balance argument, reducing optimization or computation to invariant arithmetic.
How to solve
- Replace the 1-10 / 11-20 sets with 1-12 / 13-24 (shift by 12) — pair-sum changes but the closure argument is identical
- Build a new rectilinear-polygon closure problem where one vertical side X is unknown and the up-segs = down-segs equation pins it
- Pair-with-target-sum partition: choose k items from {1..2n} such that no two sum to 2n+1; total is forced
Sub-archetype mix (4)
Click a row to see member problems.
- rectilinear-perimeter-closure 9% (3)
A closed rectilinear path partitions its boundary segments into two opposing directions (up vs down, or left vs right); the total length in each direction must be equal. Set up the equation (sum of one direction) = (sum of the other) to recover one missing segment length.
- universe-total-minus-known 41% (14)
The entire universe has a computable fixed total that can be derived from structure alone (e.g., all dice faces, all game wins, all array entries). The universe partitions into a known portion and one unknown piece; subtract the known sum from the fixed total to find the missing value.
no figureamc10-2003B-12no figureamc10-2003B-15
amc10-2005A-17no figureamc10-2008B-4no figureamc10-2011A-17no figureamc10-2012A-8no figureamc10-2014B-18no figureamc10-2015A-5no figureamc10-2016A-13no figureamc10-2016B-15no figureamc10-2016B-5
amc8-2000-8no figureamc8-2006-20no figureamc8-2009-21 - uniform-pair-sum-recovery 35% (12)
Each object pairs with exactly one partner so that every (object, partner) pair sums to the same constant S. Use this uniform pair-sum to recover unknown values, derive aggregate totals, or verify uniqueness — the structural key is that S is fixed regardless of which specific pairs form.
- forbidden-complement-selection 15% (5)
A universe partitions into (k, k+d) forbidden pairs; a valid selection must choose at most one element from each pair. When selection size equals exactly half the universe, the selected and unselected halves are forced complements, pinning the total sum or proving a bipartite structure that allows counting or strategy derivation.
More data (year-over-year, tool fingerprint, grade distribution, all members)
Tool fingerprint (1–17)
Grade distribution
- Gr 3 4
- Gr 4 5
- Gr 5 3
- Gr 6 16
- Gr 7 4
- Gr 8 2