The Fragmented Grid

Black out the boxes of this 16 × 16 grid 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	13	14	15	16
1	·	■	·	·	·	·	■	·	·	·	·	■	■	·	·	·
2	■	·	■	■	·	■	■	·	·	·	■	·	■	·	■	·
3	·	■	■	·	·	·	·	·	·	·	·	■	·	■	■	·
4	■	·	·	·	·	■	·	■	■	·	·	■	■	·	■	■
5	·	■	■	·	■	·	·	·	·	■	·	■	·	·	■	■
6	·	·	·	·	■	·	·	·	■	·	■	■	■	·	·	■
7	■	■	■	·	·	■	■	·	·	■	■	■	·	■	■	■
8	·	·	·	■	■	·	·	·	·	·	■	■	·	·	■	·
9	·	·	■	·	■	■	■	·	■	·	■	■	■	■	·	·
10	·	■	·	·	·	■	·	·	■	■	■	·	■	·	■	■
11	·	·	·	·	■	·	·	■	·	·	■	·	■	■	·	·
12	·	■	·	·	■	·	·	■	■	■	·	·	·	■	■	·
13	■	·	·	·	·	■	■	·	■	■	·	■	·	■	■	■
14	·	■	·	·	■	·	■	·	·	■	·	■	·	■	■	·
15	■	·	·	·	·	·	·	■	■	·	■	■	·	■	·	■
16	■	■	■	·	■	·	·	·	■	■	■	·	■	■	■	·

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 52 states, encountered 34 contradictions and reached 7 levels of assumptions. These measures describe this solver; they do not represent a universal ranking of players.
L7: 15 compatible placements; L7C3 black, L7C10 black, L7C14 black.
L10: 35 compatible placements; L10C1 empty, L10C3 empty.
L13: 31 compatible placements. L13C14 is filled.
C2: 5 compatible placements; L9C2 empty, L11C2 empty, L12C2 black, L13C2 empty, L14C2 black, L15C2 empty, L16C2 black.
C3: 74 compatible placements. L8C3 is empty.
C10: 62 compatible placements. L6C10 is empty.
C11: 56 compatible placements. L8C11 is filled.
C12: 20 compatible placements; L6C12 black, L7C12 black, L8C12 black, L9C12 black.
C14: 23 compatible placements; L6C14 empty, L8C14 empty, L11C14 black, L12C14 black, L14C14 black.
C15: 6 compatible placements; L2C15 black, L3C15 black, L4C15 black, L7C15 black, L12C15 black, L13C15 black.
L2: 21 compatible placements; L2C14 empty, L2C16 empty.
L7: 6 compatible placements; L7C9 empty, L7C11 black, L7C13 empty, L7C16 black.
L8: 12 compatible placements; L8C10 empty, L8C13 empty.
L9: 25 compatible placements. L9C16 is empty.
L11: 155 compatible placements. L11C16 is empty.
L12: 15 compatible placements; L12C1 empty, L12C3 empty, L12C13 empty, L12C16 empty.
L14: 20 compatible placements; L14C1 empty, L14C3 empty, L14C16 empty.
L16: 30 compatible placements. L16C3 is filled.
C1: 113 compatible placements; L15C1 black, L16C1 black.
C3: 10 compatible placements; L6C3 empty, L15C3 empty.
C13: 4 compatible placements; L1C13 black, L2C13 black, L3C13 empty, L4C13 black, L5C13 empty, L6C13 black, L9C13 black, L10C13 black, L11C13 black.
C16: 4 compatible placements; L1C16 empty, L3C16 empty, L5C16 black, L6C16 black, L10C16 black, L13C16 black.
L1: 81 compatible placements. L1C15 is empty.
L2: 15 compatible placements. L2C12 is empty.
L3: 45 compatible placements. L3C14 is filled.
L4: 20 compatible placements; L4C11 empty, L4C12 black, L4C14 empty, L4C16 black.
L5: 70 compatible placements; L5C14 empty, L5C15 black.
L6: 22 compatible placements; L6C11 black, L6C15 empty.
L10: 10 compatible placements; L10C12 empty, L10C14 empty, L10C15 black.
L11: 63 compatible placements; L11C12 empty, L11C15 empty.
L13: 11 compatible placements. L13C13 is empty.
L16: 20 compatible placements. L16C4 is empty.
C11: 21 compatible placements; L9C11 black, L10C11 black.
C12: 3 compatible placements; L1C12 black, L3C12 black, L5C12 black, L14C12 black.
C14: 2 compatible placements; L15C14 black, L16C14 black.
C15: 1 compatible placements; L8C15 black, L9C15 empty, L14C15 black, L15C15 empty, L16C15 black.
C16: 2 compatible placements. L8C16 is empty.
L1: 36 compatible investments; L1C11 empty, L1C14 empty.
L3: 9 compatible placements. L3C11 is empty.
L5: 35 compatible placements. L5C11 is empty.
L9: 10 compatible placements. L9C5 is filled.
L10: 4 compatible placements; L10C8 empty, L10C9 black, L10C10 black.
L14: 10 compatible placements; L14C11 empty, L14C13 empty.
L15: 25 compatible placements. L15C13 is empty.
C9: 17 compatible placements. L8C9 is empty.
C11: 1 compatible placements; L2C11 black, L11C11 black, L12C11 empty, L13C11 empty, L15C11 black, L16C11 black.
C13: 1 compatible placements. L16C13 is filled.
C14: 1 compatible placements. L9C14 is filled.
L2: 10 compatible placements. L2C10 is empty.
L9: 7 compatible placements. L9C10 is empty.
L11: 22 compatible placements. L11C10 is empty.
L12: 6 compatible placements; L12C8 black, L12C12 empty.
L16: 3 compatible placements; L16C8 empty, L16C9 black, L16C10 black, L16C12 empty, L16C16 empty.
C9: 12 compatible placements; L14C9 empty, L15C9 black.
C10: 3 compatible placements; L12C10 black, L13C10 black, L14C10 black, L15C10 empty.
C12: 1 compatible placements; L13C12 black, L15C12 black.
C16: 1 compatible placements. L15C16 is filled.
L12: 3 compatible placements; L12C7 empty, L12C9 black.
L13: 4 compatible placements; L13C8 empty, L13C9 black.
L15: 1 compatible placements; L15C4 empty, L15C5 empty, L15C6 empty, L15C7 empty, L15C8 black.
C7: 71 compatible placements. L16C7 is empty.
C8: 9 compatible placements; L11C8 black, L14C8 empty.
C9: 10 compatible placements; L9C9 black, L11C9 empty.
L9: 4 compatible placements. L9C8 is empty.
L11: 5 compatible placements. L11C7 is empty.
Case analysis on L16: placement 1/2 = ■^^^^··^^^^^^^··.
Case analysis on L12: placement 2/3 = * * * * * * * * * *
C5: 14 compatible placements. L13C5 is empty.
L13: 2 compatible placements; L13C6 black, L13C7 black.
C6: 35 compatible placements. L14C6 is empty.
C7: 35 compatible placements. L14C7 is filled.
Case analysis on L13: placement 1/2 =·······················································································································································································································································································································
Case analysis on L14: placement 2/2 = ■·····················································································································································································································································································································
C5: 5 compatible placements; L7C5 empty, L8C5 black, L10C5 empty, L11C5 black.
L7: 4 compatible placements; L7C2 black, L7C7 black.
L8: 2 compatible placements; L8C1 empty, L8C2 empty, L8C7 empty, L8C8 empty.
L11: 1 compatible placements; L11C1 empty, L11C3 empty, L11C4 empty, L11C6 empty.
C2: 1 compatible placements; black L1C2, empty L2C2, black L3C2, empty L4C2, black L5C2, empty L6C2.
C3: 3 compatible placements; L2C3 black, L9C3 black.
C7: 11 compatible placements. L6C7 is empty.
L1: 7 compatible placements; L1C1 empty, L1C3 empty.
L3: 2 compatible placements; L3C4 empty, L3C5 empty, L3C6 empty, L3C7 empty, L3C8 empty, L3C9 empty, L3C10 empty.
C3: 1 compatible placements; L3C3 black, L4C3 empty, L5C3 black.
C6: 8 compatible placements. L9C6 is filled.
L3: 1 compatible placements. L3C1 is empty.
L5: 10 compatible placements; L5C1 empty, L5C4 empty.
L9: 1 compatible placements; L9C1 empty, L9C4 empty, L9C7 black.
C1: 2 compatible placements; L2C1 black, L4C1 black.
C7: 2 compatible placements. L10C7 is empty.
L2: 3 compatible placements; L2C4 black, L2C5 empty.
C4: 5 compatible placements. L1C4 is empty.
C5: 2 compatible placements; L1C5 empty, L5C5 black.
L5: 4 compatible placements. L5C6 is empty.
C6: 6 compatible placements. L4C6 is filled.
L4: 2 compatible placements. L4C5 is empty.
C5: 1 compatible placements. L6C5 is filled.
L6: 3 compatible placements; L6C4 empty, L6C6 empty.
C6: 2 compatible placements; black L7C6, empty L8C6, black L10C6.
L7: 2 compatible placements. L7C8 is empty.
L8: 1 compatible placements. L8C4 is filled.
L10: 1 compatible placements. L10C4 is empty.
C4: 1 compatible placements; L4C4 empty, L7C4 empty.
L4: 1 compatible placements; L4C7 empty, L4C8 black, L4C9 black.
L7: 1 compatible placements. L7C1 is filled.
C1: 1 compatible placements. L6C1 is empty.
C7: 1 compatible placements; L1C7 black, L2C7 black, L5C7 empty.
C8: 1 compatible placements; L1C8 empty, L2C8 empty, L5C8 empty, L6C8 empty.
C9: 3 compatible placements. L5C9 is empty.
L1: 1 compatible placements; L1C6 empty, L1C9 empty, L1C10 empty.
L2: 1 compatible placements; L2C6 black, L2C9 empty.
L5: 1 compatible placements. L5C10 is filled.
L6: 1 compatible placements. L6C9 is filled.
Examples of contradictions encountered:
L7 has no legal placement left at depth 3.
C5 has no legal placement left at depth 4.
L2 has no legal placement left at depth 6.
C3 has no legal placement left at depth 6.
C5 has no legal placement left at depth 6.
L10 has no legal placement left at depth 7.
C5 has no legal placement left at depth 7.
C5 has no legal placement left at depth 7.
C1 has no legal placement left at depth 5.
L2 has no legal placement left at depth 6.
C3 has no legal placement left at depth 6.
C7 has no legal placement left at depth 6.
L10 has no legal placement left at depth 7.
C7 has no legal placement left at depth 7.
C8 has no legal placement left at depth 7.
C7 has no legal placement left at depth 4.
C3 has no legal placement left at depth 6.
C3 has no legal placement left at depth 6.
C5 has no legal placement left at depth 6.
L10 has no legal placement left at depth 7.

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": 52,
  "contradictions": 34,
  "depth": 7,
  "forced_cells": 1687,
  "branches": 17
}

The simple intersection blocks; the crosses require an analysis of nested cases (depth 7, 34 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-023

L / C	1	2	3	4	5	6	7	8	9	10	11	12	13	14	15	16
1	·	■	·	·	·	·	■	·	·	·	·	■	■	·	·	·
2	■	·	■	■	·	■	■	·	·	·	■	·	■	·	■	·
3	·	■	■	·	·	·	·	·	·	·	·	■	·	■	■	·
4	■	·	·	·	·	■	·	■	■	·	·	■	■	·	■	■
5	·	■	■	·	■	·	·	·	·	■	·	■	·	·	■	■
6	·	·	·	·	■	·	·	·	■	·	■	■	■	·	·	■
7	■	■	■	·	·	■	■	·	·	■	■	■	·	■	■	■
8	·	·	·	■	■	·	·	·	·	·	■	■	·	·	■	·
9	·	·	■	·	■	■	■	·	■	·	■	■	■	■	·	·
10	·	■	·	·	·	■	·	·	■	■	■	·	■	·	■	■
11	·	·	·	·	■	·	·	■	·	·	■	·	■	■	·	·
12	·	■	·	·	■	·	·	■	■	■	·	·	·	■	■	·
13	■	·	·	·	·	■	■	·	■	■	·	■	·	■	■	■
14	·	■	·	·	■	·	■	·	·	■	·	■	·	■	■	·
15	■	·	·	·	·	·	·	■	■	·	■	■	·	■	·	■
16	■	■	■	·	■	·	·	·	■	■	■	·	■	■	■	·

L / C	1	2	3	4	5	6	7	8	9	10	11	12	13	14	15	16
1	·	■	·	·	·	·	■	·	·	·	·	■	■	·	·	·
2	■	·	■	■	·	■	■	·	·	·	■	·	■	·	■	·
3	·	■	■	·	·	·	·	·	·	·	·	■	·	■	■	·
4	■	·	·	·	·	■	·	■	■	·	·	■	■	·	■	■
5	·	■	■	·	■	·	·	·	·	■	·	■	·	·	■	■
6	·	·	·	·	■	·	·	·	■	·	■	■	■	·	·	■
7	■	■	■	·	·	■	■	·	·	■	■	■	·	■	■	■
8	·	·	·	■	■	·	·	·	·	·	■	■	·	·	■	·
9	·	·	■	·	■	■	■	·	■	·	■	■	■	■	·	·
10	·	■	·	·	·	■	·	·	■	■	■	·	■	·	■	■
11	·	·	·	·	■	·	·	■	·	·	■	·	■	■	·	·
12	·	■	·	·	■	·	·	■	■	■	·	·	·	■	■	·
13	■	·	·	·	·	■	■	·	■	■	·	■	·	■	■	■
14	·	■	·	·	■	·	■	·	·	■	·	■	·	■	■	·
15	■	·	·	·	·	·	·	■	■	·	■	■	·	■	·	■
16	■	■	■	·	■	·	·	·	■	■	■	·	■	■	■	·

L / C	1	2	3	4	5	6	7	8	9	10	11	12	13	14	15	16
1	·	■	·	·	·	·	■	·	·	·	·	■	■	·	·	·
2	■	·	■	■	·	■	■	·	·	·	■	·	■	·	■	·
3	·	■	■	·	·	·	·	·	·	·	·	■	·	■	■	·
4	■	·	·	·	·	■	·	■	■	·	·	■	■	·	■	■
5	·	■	■	·	■	·	·	·	·	■	·	■	·	·	■	■
6	·	·	·	·	■	·	·	·	■	·	■	■	■	·	·	■
7	■	■	■	·	·	■	■	·	·	■	■	■	·	■	■	■
8	·	·	·	■	■	·	·	·	·	·	■	■	·	·	■	·
9	·	·	■	·	■	■	■	·	■	·	■	■	■	■	·	·
10	·	■	·	·	·	■	·	·	■	■	■	·	■	·	■	■
11	·	·	·	·	■	·	·	■	·	·	■	·	■	■	·	·
12	·	■	·	·	■	·	·	■	■	■	·	·	·	■	■	·
13	■	·	·	·	·	■	■	·	■	■	·	■	·	■	■	■
14	·	■	·	·	■	·	■	·	·	■	·	■	·	■	■	·
15	■	·	·	·	·	·	·	■	■	·	■	■	·	■	·	■
16	■	■	■	·	■	·	·	·	■	■	■	·	■	■	■	·