Tower Relay: use disc-count parity to choose the first move
A four-disc opening explains why the smallest disc first moves to Tower 2 in Standard, but to Tower 3 in Hard and Expert’s shortest canonical routes.
Worked examples use the Standard board unless a different size is stated. Hard and Expert may change the board dimensions or amount of information to track.
The first move depends on the whole stack
At the untouched start of Tower Relay, only disc 1 can move. Both empty towers are legal destinations, but the shortest route to Tower 3 does not begin the same way for every disc count. With an even number of discs, the smallest first moves to Tower 2. With an odd number, it first moves to Tower 3.
Standard has four discs and therefore uses the even opening. Hard’s five discs and Expert’s seven use the odd opening. This is a rule for the canonical starting stack and an optimal route, not a general instruction to apply blindly after several improvised moves. A legal detour can change which stage of the transfer you are currently performing.
Work the first six moves with four discs
Start Standard and name a move by its disc as well as its source and destination. Move disc 1 from Tower 1 to Tower 2, disc 2 from Tower 1 to Tower 3, then disc 1 from Tower 2 to Tower 3. The two smaller discs now form a legal stack on Tower 3 while discs 4 and 3 remain on Tower 1.
Next move disc 3 from Tower 1 to Tower 2, disc 1 from Tower 3 to Tower 1, then disc 2 from Tower 3 to Tower 2. Disc 1’s next move will place it onto that small stack on Tower 2. The opening has been preparing Tower 1’s largest disc to travel, rather than trying to leave every small disc on the final tower immediately.
See the smallest disc’s cycle
Along this even-disc shortest route, disc 1 visits Tower 1, then 2, then 3, then 1 again. It moves on every other turn. Between those moves, move the only legal top disc between the two towers not holding disc 1. Empty towers are allowed; a larger disc still cannot be placed on a smaller one.
For an odd starting stack aimed at Tower 3, reverse that smallest-disc cycle: Tower 1, then 3, then 2, then 1. Alternating its cycle with the available legal move builds the shortest canonical transfer. Keep track of which turn type is next; making two successive smallest-disc moves breaks the intended schedule even if both happen to be legal.
Use the pattern as a check, not an automatic repair
The parity rule is a compact way to organise a known route. The recursive explanation gives its reason: smaller stacks must be moved out of the way before the next larger disc can travel, and the direction needed by the smallest alternates as stack size changes. It is not a claim that every legal first move leads to failure when it differs from the minimum-length opening.
If you lose your place, inspect the actual stacks before continuing the cycle. Undo can return to a recognised earlier position, while Restart restores the canonical opening without changing difficulty. There is no move deadline, and a longer successful route still counts as a completion. Learn the four-disc cycle slowly before relying on it to organise the much longer seven-disc transfer.
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