PAPER DECISION STUDIES
Sliding Tiles — Decisions: compare what happens next
Four starting positions, then four separate answer sheets. The IDs match the online studies.
This version uses a four-by-four grid containing tiles 1–15 and one blank. Only a tile directly above, below, left or right of the blank can slide into it. The goal is 1–15 in reading order with the blank at R4C4. There are no diagonal slides, edge wrapping or whole-row slides. The game creates its scrambles through legal moves from the goal.
Printing and using this pack
Use Print or Save as PDF. The first page contains instructions, the next four pages contain the tasks, and the remaining pages contain answers. An answer may continue onto another page. Check your print preview before choosing a page range. Print only the challenge pages if you want to keep answers separate.
Write a move and a reason before consulting the key. Rows count from the top and columns from the left. Four in a Row and Reversi diagrams state whose turn it is. The full online comparison includes all legal first moves; the answer sheets include the complete first-move comparison or a checked route.
CHALLENGE · slide-05
A loop changes tile order
Watch the displaced tiles as you route the blank. Reach the target, then compare your route with the checked shortest route.
My move or route: __________________________________________
My reason: ________________________________________________
________________________________________________________
CHALLENGE · slide-06
Compare a longer cycle
Watch the displaced tiles as you route the blank. Reach the target, then compare your route with the checked shortest route.
My move or route: __________________________________________
My reason: ________________________________________________
________________________________________________________
CHALLENGE · slide-07
Check the odd invariant
Calculate inversions while ignoring the blank. Add the blank’s row counted from the bottom. Is this position reachable from the target? Then solve it.
My move or route: __________________________________________
My reason: ________________________________________________
________________________________________________________
CHALLENGE · slide-08
Parity survives a longer route
Calculate inversions while ignoring the blank. Add the blank’s row counted from the bottom. Is this position reachable from the target? Then solve it.
My move or route: __________________________________________
My reason: ________________________________________________
________________________________________________________
ANSWER · slide-05
A loop changes tile order
The shortest finish takes 4 moves. Moves 1–4 in the checked route circle a two-by-two block: the blank returns to R4C4 while three tiles rotate. Moving a tile out and immediately back is a two-move undo. A four-step trip around a 2×2 block instead cycles three tiles; it is not generally a reset. The distance comes from a complete breadth-first search outward from the target through 12 moves; it is not a general shortest-solution claim for arbitrary arcade shuffles.
Checked route
- Slide Tile 15 (R3C4) into the blank at R4C4.
- Slide Tile 12 (R3C3) into the blank at R3C4.
- Slide Tile 11 (R4C3) into the blank at R3C3.
- Slide Tile 15 (R4C4) into the blank at R4C3.
ANSWER · slide-06
Compare a longer cycle
The shortest finish takes 8 moves. Moves 2–5 in the checked route circle a two-by-two block: the blank returns to R3C2 while three tiles rotate. Moving a tile out and immediately back is a two-move undo. A four-step trip around a 2×2 block instead cycles three tiles; it is not generally a reset. The distance comes from a complete breadth-first search outward from the target through 12 moves; it is not a general shortest-solution claim for arbitrary arcade shuffles.
Checked route
- Slide Tile 6 (R3C2) into the blank at R2C2.
- Slide Tile 9 (R4C2) into the blank at R3C2.
- Slide Tile 14 (R4C1) into the blank at R4C2.
- Slide Tile 13 (R3C1) into the blank at R4C1.
- Slide Tile 9 (R3C2) into the blank at R3C1.
- Slide Tile 10 (R3C3) into the blank at R3C2.
- Slide Tile 11 (R4C3) into the blank at R3C3.
- Slide Tile 15 (R4C4) into the blank at R4C3.
ANSWER · slide-07
Check the odd invariant
The shortest finish takes 6 moves. There are 9 inversions and the blank is in row 4 from the bottom. Their sum 13 is odd, matching the target’s parity. Swapping two numbered tiles while keeping the blank fixed would make this sum even and that altered board impossible. The distance comes from a complete breadth-first search outward from the target through 12 moves; it is not a general shortest-solution claim for arbitrary arcade shuffles.
Checked route
- Slide Tile 1 (R2C1) into the blank at R1C1.
- Slide Tile 5 (R2C2) into the blank at R2C1.
- Slide Tile 6 (R3C2) into the blank at R2C2.
- Slide Tile 10 (R3C3) into the blank at R3C2.
- Slide Tile 11 (R3C4) into the blank at R3C3.
- Slide Tile 12 (R4C4) into the blank at R3C4.
ANSWER · slide-08
Parity survives a longer route
The shortest finish takes 10 moves. There are 14 inversions and the blank is in row 1 from the bottom. Their sum 15 is odd, matching the target’s parity. Swapping two numbered tiles while keeping the blank fixed would make this sum even and that altered board impossible. The distance comes from a complete breadth-first search outward from the target through 12 moves; it is not a general shortest-solution claim for arbitrary arcade shuffles.
Checked route
- Slide Tile 14 (R3C2) into the blank at R4C2.
- Slide Tile 6 (R2C2) into the blank at R3C2.
- Slide Tile 7 (R1C2) into the blank at R2C2.
- Slide Tile 2 (R1C3) into the blank at R1C2.
- Slide Tile 3 (R2C3) into the blank at R1C3.
- Slide Tile 7 (R2C2) into the blank at R2C3.
- Slide Tile 6 (R3C2) into the blank at R2C2.
- Slide Tile 10 (R3C3) into the blank at R3C2.
- Slide Tile 11 (R4C3) into the blank at R3C3.
- Slide Tile 15 (R4C4) into the blank at R4C3.