overlap-and-complement-counting
Two classifications partition a population (red/white × high-top/low-top) or we want 'at least one' adjacent pair. Direct counting tangles the cases; the trick is either a 2×2 contingency table (inclusion-exclusion) or counting the complementary event and subtracting from 1.
How to solve
- Change the two fractions in problem 19 (3/5 and 2/3) to (4/7 and 3/4) — table arithmetic changes, structure identical
- For problem 25 style, change rows × seats from 4×3 to 3×4 or 5×2 — the per-row adjacent count formula updates
- Try 'maximum red-high-top' instead of minimum to flip the optimization
Sub-archetype mix (6)
Click a row to see member problems.
- two-set-pie-solve 23% (21)
The universe is partitioned by two (or three) overlapping sets with known individual and union sizes; apply |A∪B| = |A|+|B|−|A∩B| to find the intersection or a specific region cell.
no figureamc10-2003A-20no figureamc10-2004B-2no figureamc10-2004B-21no figureamc10-2005B-13no figureamc10-2011A-6no figureamc10-2017B-13no figureamc10-2017B-17no figureamc10-2020B-11no figureamc10-2021B-4no figureamc10-2025A-12
amc8-1999-9no figureamc8-2000-23
amc8-2007-13no figureamc8-2007-21no figureamc8-2008-11no figureamc8-2011-6no figureamc8-2015-15no figureamc8-2019-11no figureamc8-2019-15no figureamc8-2020-23no figureamc8-2025-7 - contingency-table-optimize 12% (11)
Two binary attributes cross-classify a population into a 2×2 table whose row and column totals are given (often as fractions); find the minimum or maximum of one cell, or fill all cells from the totals.
- complement-forbidden-arrangement 28% (26)
Count valid arrangements or outcomes by computing (total − forbidden), where the forbidden set is structurally defined by an explicit rule such as adjacency, a blocked grid point, a required prefix, or graph distance.
no figureamc10-2002B-18no figureamc10-2003A-15no figureamc10-2004B-11no figureamc10-2005B-18no figureamc10-2006A-21no figureamc10-2007A-12no figureamc10-2009A-24no figureamc10-2010B-22no figureamc10-2012B-24no figureamc10-2013A-25no figureamc10-2013A-7no figureamc10-2016A-17no figureamc10-2016B-22no figureamc10-2017A-23no figureamc10-2017A-8no figureamc10-2017B-16no figureamc10-2018A-18no figureamc10-2018B-5no figureamc10-2019B-19no figureamc10-2025B-11
amc8-2003-18
amc8-2005-21no figureamc8-2008-24no figureamc8-2014-11no figureamc8-2016-17no figureamc8-2020-10 - at-least-one-complement 20% (19)
Find the probability that at least one target event occurs across draws, positions, or trials by computing 1 minus the probability that no target event occurs; applies to independent repeated trials, geometric probability, and multi-location PIE.
no figureamc10-2008B-17no figureamc10-2009B-19no figureamc10-2010A-18no figureamc10-2014B-16no figureamc10-2015B-18no figureamc10-2017B-9no figureamc10-2019A-22no figureamc10-2019B-17no figureamc10-2021B-22no figureamc10-2022B-12no figureamc10-2022B-23no figureamc8-2001-18
amc8-2009-10no figureamc8-2011-18no figureamc8-2016-13no figureamc8-2016-21
amc8-2018-23
amc8-2023-23
amc8-2024-25 - parity-partition-complement 12% (11)
Partition the outcome space by even/odd parity and use the complement 1−P(all odd) to find P(product or sum is even), or directly count the (odd, even) and (even, odd) cases when the target requires exactly one even factor.
- geometric-area-inclusion 5% (5)
Two overlapping geometric shapes or a region partitioned by a path; apply Area(union) = Area1 + Area2 − Area(intersection), or establish an area-balance equation between inside-circle/outside-square regions to find total covered area or a radius.
More data (year-over-year, tool fingerprint, grade distribution, all members)
Tool fingerprint (1–17)
Grade distribution
- Gr 2 2
- Gr 3 1
- Gr 4 13
- Gr 5 2
- Gr 6 13
- Gr 7 53
- Gr 8 8