The Hidden Border

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 comprehensive control visited 29 states, encountered 16 contradictions and reached 10 levels of assumptions. These measures describe this solver; they do not represent a universal ranking of players.
L2: 21 compatible placements; L2C6 black, L2C7 black, L2C8 black, L2C9 black.
L4: 7 compatible placements; L4C2 black, L4C11 black, L4C12 black, L4C15 black.
L6: 23 compatible placements. L6C5 is filled.
L7: 21 compatible placements. L7C6 is filled.
L8: 12 compatible placements; L8C4 black, L8C7 black, L8C8 black, L8C9 black, L8C10 black.
C2: 74 compatible placements. L5C2 is filled.
C6: 6 compatible placements; L1C6 empty, L3C6 empty, L4C6 empty, L6C6 black.
C7: 15 compatible placements; L3C7 black, L4C7 black, L7C7 empty, L9C7 empty.
C9: 35 compatible placements; L3C9 black, L4C9 black.
C10: 20 compatible placements; L6C10 black, L7C10 black, L9C10 black, L10C10 black, L11C10 black.
L1: 20 compatible placements; L1C1 black, L1C2 empty, L1C3 black, L1C4 black, L1C5 black.
L3: 24 compatible placements; L3C8 empty, L3C10 empty.
L4: 3 compatible placements; L4C8 empty, L4C10 empty, L4C13 black, L4C14 empty, L4C16 black.
L6: 12 compatible placements; L6C1 empty, L6C4 black.
L7: 4 compatible placements; L7C1 black, L7C2 black, L7C3 empty, L7C4 black, L7C5 black, L7C8 empty, L7C13 black.
L9: 78 compatible placements. L9C11 is empty.
L11: 39 compatible placements. L11C12 is empty.
C1: 8 compatible placements. L2C1 is empty.
C2: 25 compatible placements; L2C2 empty, L6C2 black.
C3: 22 compatible placements; L2C3 black, L3C3 black, L4C3 black, L5C3 empty.
C4: 11 compatible placements; black L2C4, empty L3C4, empty L4C4.
C5: 18 compatible placements; L2C5 black, L3C5 empty, L8C5 black.
C8: 24 compatible placements. L1C8 is filled.
C10: 6 compatible placements; L12C10 black, L13C10 black.
L1: 16 compatible placements. L1C9 is filled.
L2: 4 compatible placements; L2C10 black, L2C11 black, L2C12 empty.
L3: 3 compatible placements; L3C1 empty, L3C2 black, L3C14 black, L3C15 black.
L4: 1 compatible placements; L4C1 empty, L4C5 black.
L5: 70 compatible placements. L5C1 is filled.
L6: 5 compatible placements; L6C3 black, L6C7 empty, L6C8 empty.
L8: 6 compatible placements; L8C3 empty, L8C6 empty, L8C11 black.
C1: 1 compatible placements; L8C1 black, L9C1 black, L10C1 black, L11C1 empty, L12C1 empty, L13C1 empty, L14C1 empty, L15C1 empty, L16C1 empty.
C2: 15 compatible placements. L8C2 is empty.
C5: 7 compatible placements; L5C5 black, L9C5 empty.
C6: 5 compatible placements. L5C6 is filled.
C9: 23 compatible placements. L5C9 is empty.
C10: 3 compatible placements. L1C10 is empty.
C11: 15 compatible placements; L1C11 black, L3C11 black, L5C11 empty, L7C11 empty.
C12: 74 compatible placements. L1C12 is empty.
C14: 56 compatible investments; L1C14 empty, L2C14 empty.
L1: 3 compatible placements. L1C7 is filled.
L3: 2 compatible placements. L3C12 is empty.
L5: 10 compatible placements; L5C4 empty, L5C7 empty.
L7: 3 compatible placements. L7C9 is filled.
L9: 18 compatible placements. L9C2 is empty.
L10: 39 compatible placements. L10C2 is empty.
L12: 86 compatible placements. L12C9 is empty.
C4: 5 compatible placements; L9C4 black, L10C4 empty.
C12: 29 compatible placements; L5C12 black, L6C12 empty.
L6: 3 compatible placements. L6C9 is filled.
L9: 17 compatible placements; L9C3 black, L9C9 empty.
C3: 1 compatible placements; L10C3 black, L11C3 black, L12C3 empty, L13C3 empty, L14C3 empty, L15C3 empty, L16C3 empty.
Case analysis on L3 : placement 1/2 = * * * * * * * * * * *
C13: 7 compatible placements; L1C13 empty, L2C13 empty, L5C13 empty, L12C13 black, L13C13 black.
L5: 4 compatible placements. L5C15 is filled.
L12: 25 compatible placements. L12C11 is empty.
C15: 50 compatible placements. L1C15 is empty.
L1: 1 compatible placements. L1C16 is filled.
C16: 15 compatible placements; L2C16 empty, L5C16 black, L6C16 empty.
L2: 1 compatible placements. L2C15 is filled.
L5: 2 compatible placements. L5C14 is empty.
L6: 2 compatible placements. L6C14 is filled.
C14: 15 compatible placements; L7C14 black, L8C14 empty.
C15: 35 compatible placements. L6C15 is empty.
L6: 1 compatible placements. L6C13 is filled.
L7: 1 compatible placements; L7C12 empty, L7C15 empty, L7C16 black.
L8: 1 compatible placements; L8C13 empty, L8C15 black, L8C16 black.
C15: 10 compatible placements. L9C15 is empty.
C16: 10 compatible placements. L9C16 is empty.
Case analysis on L5: placement 1/2 = ■······················································································································································································································································································································
C8: 10 compatible placements; L9C8 black, L10C8 empty.
C10: 1 compatible placements; L14C10 black, L15C10 empty, L16C10 black.
L10: 15 compatible placements. L10C14 is filled.
L14: 33 compatible placements. L14C9 is empty.
C14: 4 compatible placements; L9C14 empty, L11C14 empty.
Case analysis on L9: placement 2/3 = '·················.
C6: 1 compatible placement; L10C6 black, L11C6 empty, L12C6 black, L13C6 empty, L14C6 black, L15C6 empty, L16C6 black.
C12: 6 compatible placements. L10C12 is empty.
C13: 3 compatible placements. L14C13 is filled.
L10: 4 compatible placements; L10C5 empty, L10C7 empty, L10C15 black.
L11: 8 compatible placements. L11C5 is empty.
L12: 9 compatible placements; L12C5 empty, L12C7 empty.
L14: 6 compatible placements; L14C5 empty, L14C7 empty, L14C11 empty, L14C14 black, L14C16 empty.
C14: 2 compatible placements; L12C14 empty, L16C14 empty.
C15: 6 compatible placements. L11C15 is empty.
Case analysis on C14: placement 2/2 =······················································································································································································································································································································
Case analysis on C5 : placement 1/3 = ■^^^^^^^^·············································································································································································································································································································
Case analysis on C13: placement 2/3 = * * * * * * * * * * *
L10: 2 compatible placements. L10C16 is filled.
L11: 2 compatible placements; L11C7 empty, L11C8 empty, L11C11 black, L11C16 empty.
L15: 11 compatible placements. L15C16 is empty.
C11: 3 compatible placements. L10C11 is empty.
C16: 2 compatible placements. L16C16 is filled.
L10: 1 compatible placements. L10C9 is filled.
L16: 3 compatible placements; L16C2 empty, L16C4 empty, L16C7 empty, L16C15 empty.
C2: 3 compatible placements. L14C2 is filled.
Case analysis on L11: placement 1/2 = '···················
C2: 2 compatible placements. L12C2 is empty.
Case analysis on C2 : placement 1/2 = * * * * * * * * * * * *
L13: 5 compatible placements. L13C4 is filled.
C4: 2 compatible placements. L15C4 is empty.
Case analysis on C4: placement 2/2 = ■·····················································································································································································································································································································
L12: 1 compatible placements; L12C8 black, L12C12 empty, L12C15 black, L12C16 black.
L14: 2 compatible placements. L14C8 is empty.
C8: 2 compatible placements. L13C8 is empty.
C15: 2 compatible placements. L13C15 is empty.
C16: 1 compatible placements. L13C16 is empty.
L13: 2 compatible placements; L13C7 black, L13C12 empty.
C7: 1 compatible placements. L15C7 is empty.
C12: 1 compatible placements; L14C12 black, L15C12 empty, L16C12 black.
L14: 1 compatible placements. L14C15 is empty.
L15: 3 compatible placements. L15C15 is filled.
L16: 1 compatible placements; L16C8 empty, L16C9 empty, L16C11 black.
C8: 1 compatible placements. L15C8 is filled.
C11: 1 compatible placements; L13C11 empty, L15C11 empty.
L13: 1 compatible placements. L13C9 is filled.
L15: 1 compatible placements. L15C9 is empty.
Examples of contradictions encountered:
C15 has no legal placement left at depth 4.
C16 has no legal placement left at depth 5.
C15 has no legal placement left at depth 5.
C12 has no legal placement left at depth 5.
C15 has no legal placement left at depth 4.
C16 has no legal placement left at depth 6.
L13 has no legal placement left at depth 10.
C12 has no legal placement left at depth 10.
C4 has no legal placement left at depth 8.
C8 has no legal placement left at depth 7.
C16 has no legal placement left at depth 6.
C4 has no legal placement left at depth 5.
L14 has no legal placement left at depth 5.
C14 has no legal placement left at depth 3.
L10 has no legal placement left at depth 2.
C15 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": 29,
  "contradictions": 16,
  "depth": 10,
  "forced_cells": 658,
  "branches": 12
}

The simple intersection blocks; the crosses require an intertwined case analysis (depth 10, 16 impossible branches in the solver). Editorial level based on this need for coordination, not on the only format of the grid.

← Return to puzzle SOLO-024

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	·	·	·	·	·	■	·	·	·	■	■	■	·	·	·	■