In the 1966 Doctor Who episodes The Celestial Toyroom, The Hall of Dolls, The Dancing Floor, and The Final Test, The Doctor is forced to play a game, in the minimum number of moves.   Did he do it?  If so, did he do it correctly?  Let’s find out.

Initial observation:

Over the course of four episodes of Doctor Who, in 1966, The Doctor is forced by The Toymaker to play The Trilogic Game, which is described as follows.

Toymaker: Can you remember how to play?
Doctor: I am only allowed to move one piece at a time.
Toymaker: That is right.
And you must rearrange them in the same order that they are now on point C.
Doctor: And, uh, I am not permitted to put a larger piece on a small piece?
Toymaker: Correct. And you have 1,023 moves to do it in.

Unfortunately, some of those episodes are lost.  However, in 2004, the BBC made an animated version using the original audio.

Question:

Using the animated version of the episodes as a reference, does The Doctor correctly solve The Trilogic Game? More precisely, does the board shown match the dialog and/or on-screen indicators of the current move?

Expectation:

The solution to the underlying puzzle in The Trilogic Game, also known as the Towers of Hanoi, is quite simple.  So, I suspect The Doctor could have gotten it right.

Data:

For each move there are four possibilities for correctness.  When a move is both named and shown, it is “Yes” if the shown move matches the named one,  and “NO” if the shown move doesn’t match the named move.  When a move is shown but not named, it can either be a “Valid” state or an “Invalid” state (e.g., piece 0 being on stack B).  For the sake of brevity, moves that are only shown in progress are omitted.

Below is a table listing the various moves shown during the episodes.  For each move, the move’s number, when it is named, when it is shown, the correct state for that turn, and whether the shown state is correct, are given.  The correct state was generated by a C++ implementation of the solution.

Move	Named	Shown	State	Correct
0			A: 0987654321 B: C:	Yes
23			A: 09876321 B: 5 C: 4	Yes
152			A: 09 B: 76321 C: 854	Yes
271			A: 0 B: 94321 C: 8765	Yes
444			A: 0721 B: 98 C: 6543	Yes
622			A: 432 B: 985 C: 0761	Yes
723	N/A		A: 8743 B: 96 C: 0521	Valid
770			A: 876543 B: 1 C: 092	Yes
813			A: 8743 B: 52 C: 0961	Yes
877			A: 852 B: 761 C: 0943	Yes
881			A: 81 B: 765 C: 09432	Yes
898	N/A		A: 2 B: 76543 C: 0981	Valid
900			A: 21 B: 7654 C: 0983	Yes
902			A: B: 76541 C: 09832	Yes
930			A: 61 B: 72 C: 098543	Yes
951			A: 65 B: 74 C: 098321	NO
952	N/A		A: 654 B: 7 C: 098321	Valid
976	N/A		A: 6 B: 5 C: 09874321	Valid
977	N/A		A: 61 B: 5 C: 0987432	Valid
979			A: 6 B: 521 C: 098743	NO
1000			A: 4 B: 5 C: 09876321	Yes
1022			A: B: 1 C: 098765432	Yes
1023			A: B: C: 0987654321	Yes

The following table lists moves that are mentioned, but the board isn’t shown, so correctness can’t be determined.

Move	Time
1	14:27
13	14:59
153	16:49
170	17:50
173	18:02
177	18:17
200	19:38
204	19:53
220	20:50
230	21:25
240	21:59
250	22:44
265	27:07
468	29:35
501	33:35
512	34:34
515	38:51
520	40:49
540	44:39
550	46:39
608	49:09
827	1:03:29
836	1:04:32
840	1:01:48
905	1:17:24
1021	1:28:55

Mistakes

Several mismatches occurred between the stated move and the shown board state.  Also, there are some moves that are mentioned, but not shown, so we can’t verify their correctness.  There are also some moves that are shown, but not named, those all appear to be valid states.

Moves 902 & 905:

Move 905 is shown on the screen, when The Toymaker jumps to move 930, however, the animation to update to move 930 starts at move 902. So, turn 905 is actually skipped. Move 902 is also shown on the robot when it should have read a number between 608 and 622.

Taking a closer look at the robot’s screen, there is a blurry but discernible 902.

Similarly, of the moves that aren’t shown, move 840 is shown on the screen out of order as well.

Move 951:

The mistake on move 951 appears to be a simple off by one error.  When the screen shows move 951, the board is on move 952.

Move 979:

Below are the correct states of the board after 977 and 979 moves.

After 977 moves:
	A: 61
	B: 5
	C: 0987432 

After 979 moves: 
	A: 6 
	B: 521 
	C: 098743

When move 979 is shown on the move counter, the board shows 5 by itself on B, 1 and 6 on B, with all others on C, which is move 977, not 979.

Conclusion:

Several minor mistakes were made in the depiction of the Towers of Hanoi puzzle.  The worst error was showing a state that was 2 moves off.  Other than that, the other moves were correct, not shown, or skipped by The Toymaker.  Even the mistakes were still valid board states, just not the correct ones.

Future Questions:

While some mistakes were found in the animated version of these episodes, were these mistakes made in what remains of the original 1966 episodes, or were they introduced in the animated version?