The Celestial Toymaker

A surreal piece of Dr. Who’s adventures (aren’t most of them?), with much theatricality built in. It weaves in the classic mythical story associated with the “Tower of Hanoi” puzzle.

The book opens with the Tardis and its travellers mysteriously trapped in some No-Where space by a god-like entity, the Celestial Toymaker.

“if the Doctor and his companions had been able to look outside (the scanner was not on) they would have been able to see that the TARDIS was standing in the middle of a large octagonal room, not unlike the interior of the TARDIS itself. From each angle, lines stretched to the centre of the room. The TARDIS was standing at the exact point where the lines intersected. […] Inside his ornate study, the Celestial Toymaker, the being who had captured the TARDIS and its inhabitants, was surveying his extraordinary kingdom. The Toymaker’s study appeared at first like a room. Then, as you became accustomed to its dimensions, you realised that instead of a roof there was a black immensity of outer space and the twinkling stars of the galaxies. The walls stretched up towards the blackness until they became indistinguishable from space and merged with it.”

The Celestial Toymaker is a very sinister alien, appearing as “a tall imposing figure, dressed as a Chinese mandarin with a circular black hat embossed with heavy gold thread, a large silver red and blue.”. He has had multiple prior encounters with the Doctor before, in different versions of the Doctor, though this particular encounter is with the First Doctor. In the Doctor’s own description of the Toymaker:

“Dodo obediently turned towards the Doctor. ‘Who is the Celestial Toymaker?’ she queried. ‘A powerful evil.’ The Doctor’s face darkened. ‘He has created a universe entirely in his own vision, where he manipulates people and turns them into his playthings. He gains control of your mind through these screens. Be careful, it’s a trap. […] ‘He is trying to get us in his power and make us a permanent fixture in his universe,’ said the Doctor. ‘That’s the reason for those wall screens. He’s using your mind and imagination against you. Those are memory devices that project difficult and upsetting times in your previous lives. You must fight it.’

Having captured the Doctor and his crew, the Toymaker tells them they have to win several games against the Toymaker’s animated characters before they can hope to be released. With the good Doctor, the Toymaker sets up a private game called “Trilogic”. And what is Trilogic? Here’s the excerpt:

“The Toymaker smiled. ‘At last,’ he said. ‘Here, Doctor.’ He rose, turned, and gestured over towards the table on which stood the trilogic game. He waved his hand and the three piles resolved into one big pyramid. Each segment of the pyramid from the tiny cone at the top to the largest segment at the bottom was numbered. ‘All you have to do Doctor is to reassemble these segments in the same order they are now, on point C. He pointed to one of the three triangles of the game.

Interested despite himself, the Doctor stared at the board and quickly calculated. ‘I’m only allowed to move one piece at a time right?’ he said. The Toymaker nodded. ‘That’s right.’ ‘And I’m never permitted to put a larger piece on a smaller one?’ said the Doctor, pleased as always when his sometimes fallible memory worked efficiently. ‘Absolutely correct,’ said the Toymaker. ‘And you have 1023 moves to do it in, and that is the exact amount, mind you - no more or less. If you make one mistake - you lose!’ He crossed back to the desk and pushed a lever. At the far side of the table, there was a tally recorder with two lines of figures. ‘This is to help you count. The top line shows 1023 moves. As you progress, the bottom will record the moves you make.

This is the classic “Tower of Hanoi” situation, this time with 10 plates and 3 “spindles”. The minimum number of moves to move all 10 plates from 1 place to another, following the above rules, is 2^10-1 = 1023.

The Toymaker tells the Doctor that he must not finish the game ahead of his companions, who are occupied with simpler games of snakes and ladders, cards, etc.

While Trilogic appears to be a single-person game, it has a hidden component revealed at the very end. For it turns out that if the doctor makes the final move numbers 1023, the entire world of the Toymaker, along with all its occupants like the Doctor and Tardis, would vanish. This is just as in the original ‘Tower of Hanoi’ tale, where the world is considered to be Brahma’s dream which crumbles when the last plate is moved into place on a diamond spindle by a priest in Benares…

So the Doctor is in a Catch-22 - if he does not make the final move, no one can leave the Celestial Toyroom. If he does, the world is destroyed. In a very climactic dialogue, the situation unfolds (reminded me of Wes Craven’s movie, “Wishmaster”, with its vey eerie rasping command, “Make your wish”:

“The Toymaker shrugged. ‘Make your move, Doctor,’ he said. The Doctor shook his head. ‘If I do so, then this place vanishes.’ The Toymaker nodded. ‘Yes, you will have really won.’ ‘If this place vanishes, then the TARDIS and the rest of us will vanish with it,’ said the Doctor. The Toymaker smiled a self-satisfied smile. ‘Correct. That is the price of success. Make your last move, Doctor. Make your last move.’ ”

The Doctor finds a way of getting out of this impossible situation and they all escape, leaving the Toymaker’s world in non-existence. The Celestial Toymaker, of course, is a trans-dimensional, immortal power and can continue to build new worlds to continue his game.

Evidently, the dramatization of this story with BBC is mostly lost, with only one of the 4 weekly parts surviving on tape. I recommend watching the following two clips of the ending:

https://www.youtube.com/watch?v=w9kbL8_m1qY

https://www.youtube.com/watch?v=Vi7GJ2jS8y0

Further Commentary: Alex Kasman had raised an objection to having this story categorized as mathfiction, since the presence of a game or its analysis constitutes mathfiction. I don’t disagree with him that ‘the game does not mathfiction make’, otherwise every story involving poker or chess would have to be included, for example. However, a very specific problem like “Tower of Hanoi” has a special place in my opinion because it is very niche, is commonly used in teaching a very specific mathematical technique like induction, and such stories are not very common. So when I first read this story, there was an automatic bulb of “mathfiction” which went off in my mind and I did not even hazard a second guess till Alex raised his (very valid) objection.

So I suppose there is some amount of artistic license of judgment involved in categorizing this as mathfiction, and do believe that inclusion helps a math discussion for those who come across it.