Tower of London Strategy: Counting Moves to Hit Par Every Time

Updated October 2026

Tower of London is a planning test from neuropsychology, rebuilt as a game against the clock. You rearrange coloured balls on three pegs until they match a target picture, and you must do it in exactly the minimum number of moves. Trial and error is not allowed. This guide explains how the puzzles are built, how the clock and the par rise as you go, and a counting method that lets you check a plan before you touch the screen.

The rules in brief

The small picture at the top is the goal and the large one below is your board. Tap a peg to lift its top ball and tap another peg with free space to set it down. Each peg holds only as many balls as its height. Lifting a ball and tapping the same peg again puts it back without costing a move, and tapping a full peg while holding a ball does nothing at all. Each ball has a shape as well as a colour (circle, triangle, square, diamond and star), so you can read the puzzle by shape if colours are hard to tell apart.

The number shown as par is the true minimum, and the game finds it by searching every reachable position. If your board matches the goal when your move count reaches par, you score the par in points. If you reach par without matching it, the attempt fails at that moment and you lose one of three lives. Running out of time also costs a life. A failed puzzle is not repeated. The game moves on to the next problem, with the same rise in par and drop in time as if you had solved it.

The three stages

The puzzles come in three sizes. Problems 1 to 8 use three balls on pegs that hold 3, 2 and 1. Problems 9 to 18 use four balls on pegs of 4, 3 and 2. From problem 19 onward there are five balls on pegs of 5, 4 and 3. These boards are small enough to search completely: the three-ball board has 36 possible arrangements, the four-ball board 264 and the five-ball board 2,040. Within each stage, the par climbs on a fixed schedule.

• Three balls: par 2, 2, 3, 4, 5, 6, 6, 7. The hardest possible three-ball puzzle needs 8 moves.
• Four balls: par 4, 4, 5, 6, 6, 7, 8, 8, 9, 10. Ten is the four-ball maximum.
• Five balls: par starts at 6 and rises by one every second problem, reaching 12 at problem 31 and staying there. A few five-ball arrangements need 13 moves, but they are so rare that the game stops at 12.

Note the drop in par each time a stage begins. Problem 9 asks for only 4 moves after problem 8 asked for 7. That breather matters, because you are now reading a fourth ball and a taller set of pegs.

How fast the clock shrinks

The first problem gives you 45 seconds, and each later problem gets 95 percent of the one before. There is no lower limit, so the time keeps shrinking forever and every run must eventually end. Here is what that means per move of par:

• Problem 1: 45 seconds for 2 moves, more than 20 seconds per move.
• Problem 8: about 31 seconds for 7 moves, about 4.5 seconds per move.
• Problem 18: about 19 seconds for 10 moves, under 2 seconds per move.
• Problem 19, the first five-ball puzzle: about 18 seconds, roughly 40 percent of the starting allowance.
• Problem 31: under 10 seconds for 12 moves.
• Problem 40: about 6 seconds for 12 moves.

The time bar turns its warning colour in the last quarter of each allowance. There is a short pause between problems while the result is shown, and the clock does not run during it. The timer measures real time, so leaving the app does not pause it. If you go to another app mid-puzzle, expect to come back to a lost life.

Counting a plan before you move

The most useful tool for this game is a simple count. Call a ball settled if it already sits on its goal peg at its goal height, with every ball beneath it also correct. A settled ball never needs to move. Every other ball has to move at least once. So the number of unsettled balls is a lower bound on the solution, and par minus that number tells you how many extra moves your plan must contain. An extra move is one where a ball goes somewhere temporary and has to move again later.

We checked this against every pair of positions on the three-ball and four-ball boards, and a sample of the five-ball board. The bound held every time, and the size of the gap is very predictable. On three balls, a par of 5 always means exactly two extra moves. On four balls, a par of 8 means four extra moves in 95 percent of cases. Once you know how many temporary moves you are allowed, most candidate plans fail the count before you finish thinking them through. If your idea parks three balls and the budget is two, throw it out.

The count also catches wasted moves. In our search, a shortest solution almost never needs to touch a settled ball. The only exception was a peg that is already full and already correct. Then you sometimes have to borrow it as parking space and rebuild it afterwards. That happened in about 4 percent of three-ball position pairs and 0.3 percent of four-ball ones. If a plan disturbs a settled ball for any other reason, look for a shorter one.

A routine for each puzzle

1. Read the goal from the bottom up. For each goal peg, find the lowest ball that is not yet correct. Those balls are your real targets; everything above them on the goal is decided later.
2. Find what is in the way. Every ball sitting on top of a target, or occupying the spot a target needs, will have to move out. Ask where it can go that will not block something else.
3. Budget the temporary moves. Count unsettled balls, subtract from par, and make sure your plan uses exactly that many temporary moves.
4. Watch the capacities. The shortest peg fills quickly. A ball parked there must be able to leave before you need the space for something else.
5. Then play it in one go. Solving in your head is quicker than solving with your fingers, and the free lift-and-replace is a safe way to point at a ball while you think.

Scoring and how runs end

Your score is the total par of every puzzle you solve, so later puzzles are worth more. Solving all eight three-ball problems is worth 35 points. Clearing every problem through 18 adds up to 102, and a run with no mistakes at all through problem 31 would hold 216. A failed puzzle loses only its own par, but it also uses up one of three lives for the rest of the run. Since the next problem arrives harder and shorter either way, a careless early slip leaves you with less room late in the run, when the time per move is lowest.

A good habit is to spend most of your effort early. The three-ball puzzles have huge time margins, so use them to practise the counting routine until it becomes automatic. Once the five-ball stage arrives, there is no time to learn it on the clock.

Your best score is kept on your device, and the end-of-run card lets you send your score to the Hall of Fame.