Crossing Intervals

Black out the boxes of this grid 12 × 12 to satisfy all series.

This page contains the solution.

The answer

The grid below is the only solution; ■ = black, · = empty.

Coordinates: lines from top to bottom, columns from left to right.
L / C	1	2	3	4	5	6	7	8	9	10	11	12
1	■	·	■	■	■	■	■	·	·	■	·	·
2	·	■	·	■	■	·	·	·	■	·	■	■
3	■	·	·	·	■	·	■	·	·	·	·	·
4	■	■	■	·	·	■	·	■	·	■	■	■
5	■	■	■	·	·	■	■	·	■	·	■	■
6	·	·	·	■	·	■	·	·	·	·	■	·
7	·	■	■	■	·	·	·	■	·	■	·	■
8	·	·	·	·	■	·	■	■	■	·	·	·
9	■	·	■	■	■	■	■	·	·	·	·	·
10	■	·	·	■	■	·	■	·	■	·	·	■
11	■	■	·	■	·	■	·	·	·	■	■	·
12	·	■	■	·	·	■	·	■	■	·	·	■

The reasoning

Start by listing the investments in each series. Keep only those compatible with the already fixed crosses: any box common to all investments can be fixed.
This intersecting closure is not enough here. Case analyses should be conducted on a line or column, each choice should be passed on to the perpendicular lines and a choice should be rejected as soon as a series can no longer be placed.
The exhaustive control visited 9 states, encountered 6 contradictions and reached 2 levels of assumptions. These measures describe this solver; they do not represent a universal ranking of players.
L1: 20 compatible placements; L1C6 black, L1C7 black.
L4: 5 compatible placements; Black L4C2, Black L4C3, Black L4C10, Black L4C11.
L5: 5 compatible placements; L5C2 black, L5C3 black, L5C6 black, L5C11 black.
C2: 12 compatible placements; L3C2 empty, L6C2 empty.
C3: 8 compatible placements; L3C3 empty, L6C3 empty.
C6: 10 compatible placements. L2C6 is empty.
C7: 20 compatible placements. L2C7 is empty.
C10: 40 compatible placements; L3C10 empty, L5C10 empty.
L2: 9 compatible placements; L2C4 black, L2C11 black.
L5: 4 compatible placements. L5C12 is filled.
C11: 5 compatible placements; L1C11 empty, L3C11 empty, L6C11 black, L7C11 empty.
L6: 16 compatible investments; L6C10 empty, L6C12 empty.
C12: 8 compatible placements; L3C12 empty, black L4C12.
L4: 4 compatible placements. L4C9 is empty.
Case analysis on L4 : placement 3/4 = ■^^··^^^^^^.
C4: 12 compatible placements. L10C4 is filled.
Case analysis on L5 : placement 3/4 = ■^^^^^^^^^^^.
C4: 6 compatible placements; L7C4 black, L11C4 black.
C7: 7 compatible placements. L6C7 is empty.
C8: 17 compatible placements. L2C8 is empty.
C9: 22 compatible placements. L6C9 is empty.
L2: 3 compatible placements; L2C9 black, L2C10 empty, L2C12 black.
L7: 11 compatible placements. L7C1 is empty.
C1: 9 compatible placements. L10C1 is filled.
C9: 4 compatible placements; L1C9 empty, L3C9 empty.
C12: 4 compatible placements. L1C12 is empty.
L1: 3 compatible placements; L1C4 black, L1C5 black, L1C10 black.
L10: 14 compatible placements. L10C2 is empty.
C2: 6 compatible placements; black L11C2, black L12C2.
C4: 3 compatible placements. L3C4 is empty.
C5: 5 compatible placements; L2C5 black, L3C5 black.
L2: 2 compatible placements. L2C3 is empty.
L3: 2 compatible placements; L3C1 black, L3C6 empty.
L11: 10 compatible placements; L11C1 black, L11C3 empty, L11C5 empty.
C1: 2 compatible placements; black L1C1, empty L2C1, empty L6C1, empty L8C1.
C3: 3 compatible placements; L1C3 black, L12C3 black.
C5: 3 compatible placements; L8C5 black, L12C5 empty.
C6: 3 compatible placements; black L6C6, empty L7C6, black L11C6.
L1: 1 compatible placements; L1C2 empty, L1C8 empty.
L2: 1 compatible placements. L2C2 is filled.
L6: 2 compatible placements. L6C5 is empty.
L7: 6 compatible placements; L7C3 black, L7C12 black.
L11: 4 compatible placements. L11C7 is empty.
L12: 4 compatible placements; L12C1 empty, L12C4 empty, L12C9 black.
C1: 1 compatible placements. L9C1 is filled.
C3: 2 compatible placements. L8C3 is empty.
C4: 1 compatible placements; L6C4 black, L8C4 empty, L9C4 black.
C5: 2 compatible placements. L9C5 is filled.
C7: 2 compatible placements; L3C7 black, L8C7 black, L9C7 black, L12C7 empty.
C8: 10 compatible placements. L3C8 is empty.
C9: 3 compatible placements. L11C9 is empty.
C12: 3 compatible placements. L8C12 is empty.
L6: 1 compatible placements. L6C8 is empty.
L8: 2 compatible placements; L8C10 empty, L8C11 empty.
L9: 2 compatible placements; L9C2 empty, L9C6 black, L9C9 empty, L9C10 empty, L9C11 empty, L9C12 empty.
L11: 2 compatible placements; L11C8 empty, L11C11 black.
L12: 3 compatible placements. L12C6 is filled.
C6: 1 compatible placements; L8C6 empty, L10C6 empty.
C9: 2 compatible placements. L10C9 is filled.
C11: 1 compatible placements; L10C11 empty, L12C11 empty.
C12: 1 compatible placements; L10C12 black, L11C12 empty, L12C12 black.
L8: 1 compatible placements; L8C2 empty, L8C8 black, L8C9 black.
L10: 2 compatible placements; L10C7 black, L10C8 empty, L10C10 empty.
L11: 1 compatible placements. L11C10 is filled.
C2: 1 compatible placements. L7C2 is filled.
C7: 1 compatible placements. L7C7 is empty.
C8: 2 compatible placements. L12C8 is filled.
C9: 1 compatible placements. L7C9 is empty.
C10: 1 compatible placements; L7C10 black, L12C10 empty.
L7: 1 compatible placements; L7C5 empty, L7C8 black.
C5: 1 compatible placements. L10C5 is filled.
C8: 1 compatible placements. L9C8 is empty.
L9: 1 compatible placements. L9C3 is filled.
L10: 1 compatible placements. L10C3 is empty.
Examples of contradictions encountered:
C7 has no legal placement left at depth 1.
L2 has no legal placement left at depth 1.
C5 has no legal placement left at depth 2.
C5 has no legal placement left at depth 2.
C3 has no legal placement left at depth 2.
L12 has no legal placement left at depth 1.

Common mistakes and false leads

Do not confuse an unknown box with an empty proven box.
A hypothesis must be cancelled with all its consequences if a series becomes impossible.
The form suggested by the first black boxes is not evidence.

How we verified this answer

Enumeration of placements by line, propagation by intersections and exhaustive search for up to two solutions; new verification of response series.

{
  "nodes": 9,
  "contradictions": 6,
  "depth": 2,
  "forced_cells": 366,
  "branches": 2
}

The simple intersection blocks; the crosses require an analysis of nested cases (depth 2, 6 branches impossible in the solver). Editorial level based on this need for coordination, not on the only format of the grid.

← Return to puzzle SOLO-021

L / C	1	2	3	4	5	6	7	8	9	10	11	12
1	■	·	■	■	■	■	■	·	·	■	·	·
2	·	■	·	■	■	·	·	·	■	·	■	■
3	■	·	·	·	■	·	■	·	·	·	·	·
4	■	■	■	·	·	■	·	■	·	■	■	■
5	■	■	■	·	·	■	■	·	■	·	■	■
6	·	·	·	■	·	■	·	·	·	·	■	·
7	·	■	■	■	·	·	·	■	·	■	·	■
8	·	·	·	·	■	·	■	■	■	·	·	·
9	■	·	■	■	■	■	■	·	·	·	·	·
10	■	·	·	■	■	·	■	·	■	·	·	■
11	■	■	·	■	·	■	·	·	·	■	■	·
12	·	■	■	·	·	■	·	■	■	·	·	■

L / C	1	2	3	4	5	6	7	8	9	10	11	12
1	■	·	■	■	■	■	■	·	·	■	·	·
2	·	■	·	■	■	·	·	·	■	·	■	■
3	■	·	·	·	■	·	■	·	·	·	·	·
4	■	■	■	·	·	■	·	■	·	■	■	■
5	■	■	■	·	·	■	■	·	■	·	■	■
6	·	·	·	■	·	■	·	·	·	·	■	·
7	·	■	■	■	·	·	·	■	·	■	·	■
8	·	·	·	·	■	·	■	■	■	·	·	·
9	■	·	■	■	■	■	■	·	·	·	·	·
10	■	·	·	■	■	·	■	·	■	·	·	■
11	■	■	·	■	·	■	·	·	·	■	■	·
12	·	■	■	·	·	■	·	■	■	·	·	■

L / C	1	2	3	4	5	6	7	8	9	10	11	12
1	■	·	■	■	■	■	■	·	·	■	·	·
2	·	■	·	■	■	·	·	·	■	·	■	■
3	■	·	·	·	■	·	■	·	·	·	·	·
4	■	■	■	·	·	■	·	■	·	■	■	■
5	■	■	■	·	·	■	■	·	■	·	■	■
6	·	·	·	■	·	■	·	·	·	·	■	·
7	·	■	■	■	·	·	·	■	·	■	·	■
8	·	·	·	·	■	·	■	■	■	·	·	·
9	■	·	■	■	■	■	■	·	·	·	·	·
10	■	·	·	■	■	·	■	·	■	·	·	■
11	■	■	·	■	·	■	·	·	·	■	■	·
12	·	■	■	·	·	■	·	■	■	·	·	■