casework-by-position-type
Positions in a configuration (squares on a grid, vertices of a polyhedron) split naturally into a small number of symmetry classes (corner / edge / center; edge-neighbor / face-diagonal / space-diagonal). Counting attacks each class separately and sums via the multiplication principle.
How to solve
- Move from 3x3 chess grid to 4x4 — corner/edge/middle counts shift but the casework strategy is identical
- Replace kings with knights (different attack pattern) so the per-class counts change
- On the cube, ask for equilateral triangles NOT containing P, or right triangles containing P
Sub-archetype mix (5)
Click a row to see member problems.
- grid-cell-position 22% (23)
A 2D grid or 3D solid is divided into cells (or cubes) that fall into structurally distinct classes based on how many boundary faces they touch (corner, edge, face-center, interior). Casework runs over these position classes and sums the count for each.
no figureamc10-2002A-18
amc10-2004A-5no figureamc10-2012A-20no figureamc10-2018A-20no figureamc10-2021A-25no figureamc10-2023B-10no figureamc12-2004A-13
amc12-2009B-25no figureamc12-2012A-15no figureamc12-2018A-15
amc12-2020A-6no figureamc12-2023B-5
amc8-1999-20
amc8-2003-13
amc8-2005-3no figureamc8-2006-18
amc8-2008-19no figureamc8-2014-19
amc8-2015-12
amc8-2020-9
amc8-2024-17
amc8-2024-20no figuresm-12 - digit-place-position 7% (7)
A multi-digit number is analyzed by splitting cases on which digit position (units, tens, hundreds, or a rank like median/maximum) holds the constrained value. Each possible slot is one structural case.
- vertex-or-element-role 23% (24)
A small fixed set of elements (vertices of a shape, people in a group, points in a figure) is divided by which one plays a constrained structural role (right-angle vertex, driver, symmetric anchor). Casework exhausts the possible role-holders.
no figureamc10-2009B-25
amc10-2013B-22no figureamc10-2016A-18
amc10-2017B-18no figureamc10-2020A-23no figureamc10-2020B-8no figureamc10-2025A-3no figureamc12-2007A-15no figureamc12-2009B-17no figureamc12-2010B-15no figureamc12-2012B-20no figureamc12-2013A-12no figureamc12-2016A-14 amc12-2017B-13no figureamc12-2019B-21no figureamc12-2020A-20no figureamc12-2021FB-10no figureamc12-2021FB-3
amc12-2022B-24
amc8-2001-23no figureamc8-2003-16
amc8-2009-20no figureamc8-2018-11
amc8-2019-6 - count-of-type 38% (39)
A configuration is split into cases by how many elements of a given kind appear (0, 1, 2, … up to a maximum). Each possible count defines one structurally distinct case, and the totals are summed.
amc10-2004A-16no figureamc10-2004B-23no figureamc10-2007B-22no figureamc10-2010A-22no figureamc10-2011A-21no figureamc10-2012A-23
amc10-2012B-25no figureamc10-2013B-12no figureamc10-2017A-25no figureamc10-2020B-17no figureamc10-2024B-17no figureamc12-2003A-20
amc12-2003B-10no figureamc12-2004B-20no figureamc12-2005A-25no figureamc12-2005B-25no figureamc12-2006A-25no figureamc12-2007B-16no figureamc12-2009A-21no figureamc12-2009B-20no figureamc12-2010A-25no figureamc12-2012A-19no figureamc12-2012A-22 amc12-2012B-22no figureamc12-2013A-15no figureamc12-2014A-13no figureamc12-2015A-11no figureamc12-2015B-17no figureamc12-2019B-11no figureamc12-2020B-15no figureamc12-2020B-20no figureamc12-2021A-15
amc12-2022A-19no figureamc12-2023B-23no figureamc12-2023B-6no figureamc12-2023B-8no figureamc8-2000-21no figureamc8-2002-21
amc8-2011-19 - arrangement-block-type 10% (10)
Objects of two (or more) types are arranged in a line, circle, or grid, and the arrangement is classified by the pattern of consecutive same-type blocks (LLLRRR vs LRRRLL vs LRRLLR, etc.). Casework runs over the distinct block-sequence shapes.
More data (year-over-year, tool fingerprint, grade distribution, all members)
Tool fingerprint (1–17)
Grade distribution
- Gr 3 4
- Gr 4 11
- Gr 5 3
- Gr 6 5
- Gr 7 23
- Gr 8 8