Towers of London

What you see

Each trial is one puzzle. On the left, the Goal shows the arrangement you must build, drawn smaller. On the right, Your board shows the current position. There are always three pegs, numbered 1, 2 and 3 on the front of your board's base, and each has a fixed capacity, shown by its height.

• Discs are numbered 1 to 5 and each has its own colour, so you never have to rely on colour alone.
• Moving: choose a peg to lift its top disc (it rises above the peg), then choose another peg to place it there. Pegs with room show a dashed ring where the disc would land. Choose the same peg again to put the disc back. If the peg you choose is full, the disc goes back. On a keyboard, keys 1, 2 and 3 choose pegs and Escape puts a lifted disc back.
• Only the top disc moves, one disc per move, and a peg can never hold more discs than its capacity. Unlike the Tower of Hanoi, there is no size rule: any disc may sit on any other.
• Status line: one line under the board shows the moves used against the move limit, for example Moves 2 of 6, and on levels 1 to 3 the minimum as well (Shortest 4). When your board matches the goal it reads Solved in 5 moves; when the limit is used up it reads No moves left.
• Undo (U) takes back the last move and Restart (R) resets the board. When the goal is reached you can still take moves back before you submit. When the move limit is used up, no further moves are allowed until you undo or restart.

A practice session opens at difficulty 3 and adapts as you go. The side panel shows your difficulty out of 10, the trial number, your correct count and a running clock. There is no countdown on this game, but your time is recorded. Press Enter to check your answer.

How the levels change

The generator picks a goal first, then a start position at exactly the target distance, checked by a full search of every reachable position. If it cannot find a puzzle with all the extra rules, it relaxes them in order, and it may step the minimum down by up to two moves. A "trap" means that every optimal first move lifts a disc off the peg where it must finish. Five-disc puzzles use taller pegs (4, 3, 1) because on the classic board five discs leave only one free slot, and nothing deep can be built.

What it measures

Towers of London feeds the look-ahead facet of your planning score and carries the largest weight on the planning shelf, 40%. In our game's own definition it draws mainly on planning and sequencing, with a smaller working-memory component.

The task descends directly from Tim Shallice's Tower of London, published in 1982 to study planning after frontal-lobe damage. His apparatus used three coloured balls on three pegs holding three, two and one ball, exactly our classic board, which has 36 possible positions. Shallice derived it from ideas in artificial intelligence as "a task heavily loaded on planning abilities". It is a cousin of the Tower of Hanoi, the 1883 puzzle in which discs must never sit on smaller discs and three discs need at least seven moves.

Research on the task explains what our levels manipulate:

• Search depth. Puzzles needing more moves demand more mental simulation. In a balanced research problem set, difficulty rose almost perfectly linearly with minimum moves.
• Structure, not just length. Problem difficulty also depends on how clearly the goal implies an order of placement, how the start is stacked, and how many optimal routes exist. That is why levels 1 and 2 guarantee two optimal routes (more forgiving) and level 10 allows only one.
• Subgoals and traps. Errors and preparation time rise with the number of subgoal chunks, moves that are necessary but do not place a disc in its final spot. The trap levels force at least one such move right at the start.
• Two kinds of thinking. Brain imaging suggests the left dorsolateral prefrontal cortex works harder when the goal's order is ambiguous and the right when search depth increases.

How it is scored here

Only the moves you finally submit count. Moves you undo, or wipe with Restart, never reach the scorer. Moves after the goal is reached are ignored.

• Solved in the minimum: marked Correct, full credit.
• Solved in more moves: marked Partly correct, with credit equal to the minimum divided by the moves you used. A 5-move puzzle solved in 7 moves earns 5 ÷ 7, about 71%.
• Not solved: marked Incorrect. You earn partial credit below 40%, in proportion to how much closer to the goal you got at your best moment, measured in true moves-to-goal rather than by eye.

After each trial the verdict line gives the reason, for example "Partly correct · Solved in 7 moves; the shortest takes 5" or "Incorrect · Not solved within 6 moves; the closest position was 1 move from the goal". In an adaptive session, a Correct answer moves you up one level, Partly correct keeps you where you are, and Incorrect moves you down one level. So only optimal solutions climb.

For your game score, each trial's credit is combined with a small time component (15% of the weight). Your expected thinking time grows with the puzzle's difficulty, about four seconds per difficulty point, so a long pause on a hard puzzle costs you very little. Difficulty is measured from the puzzle itself: the minimum moves, plus 0.6 for each disc above three, plus one for a trap first move, minus a little when the move limit is generous. On our scale, 50 means you are clearing puzzles at the rate their difficulty predicts; above 50, you are doing better than that.

Your session report adds four figures: moves beyond the minimum, move efficiency, puzzles solved, and, where your timing data allows, thinking before moving, the ratio of time before your first move to time spent moving.

Where you will meet it

Tower tasks are standard in neuropsychology. The Cambridge computerised battery (CANTAB) includes a variant called Stockings of Cambridge, and Owen and colleagues' 1990 computerised tower study separated thinking time before the first move from thinking time during the solution.

In hiring, prep sites report that the pymetrics suite (now part of Harver) includes a tower game based on the Tower of London; the vendor does not publish a game-by-game list, so treat that as candidate-reported. Other vendors use different puzzles for the same ability; Aon, for example, describes its motionChallenge as measuring "complex planning capability". Our game is the same family of task as these, not a copy of any of them. The skills also carry straight into paper tests: seating arrangements and scheduling sets reward the same "what must be true first?" thinking, as the where planning is tested lesson shows.

The method

1. Read the goal from the bottom up

   For each goal peg, note which disc sits at the bottom. Bottom discs must be placed first, so they define your first real subgoal. Say the goal in one line, for example "peg 1: 1 then 2; peg 2: 3".

2. Check what is already settled

   A disc is settled when it sits on its goal peg at its goal height and every disc beneath it is settled too. Leave settled discs alone unless something must go underneath them. A disc on the right peg at the wrong height is not settled.

3. Find the blocker

   Pick the first empty goal slot, starting from the bottom, and ask what stops you filling it. Usually it is a disc on top of the disc you need, or a disc sitting in the slot.

4. Choose the parking spot

   Blockers need somewhere to wait. The one-disc peg is the most valuable parking spot on the board and the easiest to waste. Park a disc there only if it can stay until it is needed or can leave in one move.

5. Ask whether the first move is a backward one

   If no legal move puts a disc straight into its final place, the right move probably takes a disc away from its goal peg. On levels 7, 8 and 10 this is guaranteed by design.

6. Count before you touch

   Count the moves in your plan. On levels 1 to 3 compare with the shortest count under the board. From level 4 the minimum is hidden, but the move limit gives it away: the limit is always the minimum plus a fixed slack, 2 at levels 4 to 6 and 1 at levels 7 to 10. If your plan is longer than that, look again before you move.

7. Execute cleanly, and use Undo without guilt

   Carry out the plan. If you slip, Undo removes the move from your answer entirely. It costs only time.

Worked example 1: the disc that is on the right peg too early

Show the solution

Read the goal from the bottom: peg 1 keeps disc 1, peg 2 needs 3 at the bottom with 2 on top, and peg 3 ends empty.

Disc 2 is already on peg 2, which is tempting to leave alone, but disc 3 must go underneath it. So disc 2 has to leave its own goal peg first: a backward move.

1. Disc 2 to peg 1 (on top of disc 1; peg 3 is full, so peg 1 is the only parking spot).
2. Disc 3 to peg 2.
3. Disc 2 to peg 2.

Three moves, the minimum. Starting with disc 3 feels natural, because disc 3 is the disc that must move to peg 2, but it costs extra moves either way: parked on disc 1 it leads to a 4-move solution, and dropped onto disc 2 it builds peg 2 upside down and needs 5.

Worked example 2: a five-move puzzle with one route

Show the solution

Goal, bottom up: peg 1 needs disc 1 at the bottom and disc 2 on top; peg 2 needs disc 3; peg 3 ends empty.

Blocker: the bottom of peg 1 should hold disc 1, but disc 3 is there, with disc 1 on top of it. Disc 3 has to get out, and it cannot go to peg 2 yet because disc 2 is sitting where disc 3 finally belongs. So disc 3 needs a temporary parking spot: the one-disc peg 3.

Chunk 1 (clear the base): disc 1 to peg 2 (on top of disc 2), then disc 3 to peg 3.

Chunk 2 (rebuild): disc 1 to peg 1, disc 2 to peg 1, disc 3 to peg 2.

Five moves, and this puzzle has exactly one optimal route. The common mistake is to move disc 1 to peg 3 first, because peg 3 looks free. That fills the only parking spot that disc 3 needs, and the best you can then do is six moves.

Tips and traps

Levelling up

As the levels rise, three things get harder at once. The puzzles get deeper (up to nine moves), the move limit shrinks from three spare moves to one, and from level 7 the first move is usually a trap. At levels 9 and 10 a fifth disc arrives with taller pegs holding four, three and one.

• Levels 1 to 3: use the shortest count under the board to calibrate. Aim to match it every time; a Partly correct answer does not move you up.
• Levels 4 to 6: the minimum is hidden. Work it out from the move limit (minus 2), and plan in two-subgoal chunks.
• Levels 7 and 8: assume the first move is backward and look for which disc must step off its goal peg.
• Levels 9 and 10: with three free slots instead of two, there are more places to park, and more ways to waste moves. Decide every disc's parking spot before the first move.

Worked example 3: four discs and a trap

Show the solution

Goal, bottom up: peg 1 needs 1, then 2, then 4. Peg 2 needs 3. Peg 3 ends empty.

Disc 1 is settled. Disc 2 needs to sit on disc 1, but disc 4 is in that slot and disc 2 is buried under disc 3 on peg 2. Disc 3 belongs on peg 2 but at the bottom, where disc 2 is.

The only optimal first move is disc 3 onto peg 1, which lifts disc 3 off its own goal peg: the trap. It frees disc 2, which goes to the one-disc peg. Then disc 3 returns to the empty peg 2, disc 4 joins it temporarily, and peg 1 is rebuilt.

1. Disc 3 to peg 1.
2. Disc 2 to peg 3.
3. Disc 3 to peg 2.
4. Disc 4 to peg 2.
5. Disc 2 to peg 1.
6. Disc 4 to peg 1.

Six moves, the minimum, and this puzzle has only one optimal route. At level 7 the move limit would be seven, so there is room for just one slip.

1. Week 1
   • Three sessions of 10 minutes
   • Say the goal bottom-up before each first move
   • Target: match the shortest count on every level 1 to 3 puzzle
2. Week 2
   • Three sessions
   • Before each first move, write the order of placement
   • Target: no more than one move beyond the minimum per puzzle at levels 4 to 6
3. Week 3
   • Three sessions
   • At level 7 and above, find the backward first move before touching anything
   • Review moves beyond the minimum in your report after each session
4. Week 4
   • Four shorter sessions on consecutive days
   • Plan in two-subgoal chunks on 8- and 9-move puzzles
   • Compare your thinking-before-moving ratio with week 1

Check yourself

5 practice questions

1. At level 8, the line under the board reads Moves 0 of 8. What is the minimum number of moves for this puzzle?
2. You solve a 6-move puzzle in 8 moves after undoing two other moves along the way. What credit does the trial earn?
3. In an adaptive session, what happens to your level after a Partly correct answer?

and 2 more

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Sources

1. Shallice, T. (1982). Specific impairments of planning. Philosophical Transactions of the Royal Society of London B, 298(1089), 199–209. doi:10.1098/rstb.1982.0082
2. Millisecond Software. Tower of London task (ToL): apparatus with three pegs holding 1, 2 and 3 discs. millisecond.com; Tower of London test overview and versions. Wikipedia
3. Tower of Hanoi: history, rules and the 2^n − 1 minimum. Wikipedia
4. Kaller, C. P., Unterrainer, J. M., & Stahl, C. (2012). Assessing planning ability with the Tower of London task: Psychometric properties of a structurally balanced problem set. Psychological Assessment, 24(1), 46–53. PubMed 21859218
5. Berg, W. K., Byrd, D. L., McNamara, J. P., & Case, K. (2010). Deconstructing the tower: Parameters and predictors of problem difficulty on the Tower of London task. Brain and Cognition, 72(3), 472–482. PubMed 20167413
6. Kaller, C. P., Unterrainer, J. M., Rahm, B., & Halsband, U. (2004). The impact of problem structure on planning: Insights from the Tower of London task. Cognitive Brain Research, 20(3), 462–472. PubMed 15268923
7. Ward, G., & Allport, A. (1997). Planning and problem-solving using the five-disc Tower of London task. Quarterly Journal of Experimental Psychology, 50A(1), 49–78. doi:10.1080/713755681
8. Kaller, C. P., Rahm, B., Spreer, J., Weiller, C., & Unterrainer, J. M. (2011). Dissociable contributions of left and right dorsolateral prefrontal cortex in planning. Cerebral Cortex, 21(2), 307–317. PubMed 20522540
9. Unterrainer, J. M., Rahm, B., Kaller, C. P., et al. (2004). Planning abilities and the Tower of London: Is this task measuring a discrete cognitive function? Journal of Clinical and Experimental Neuropsychology, 26(6), 846–856. PubMed 15370380
10. Owen, A. M., Downes, J. J., Sahakian, B. J., Polkey, C. E., & Robbins, T. W. (1990). Planning and spatial working memory following frontal lobe lesions in man. Neuropsychologia, 28(10), 1021–1034. PubMed 2267054
11. JobTestPrep (prep site). pymetrics Tower game. jobtestprep.com
12. cut-e (now Aon). smartPredict product page, archived 10 December 2019. Wayback Machine

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