Tantalizer 4: In the arena
From New Scientist #553, 13th July 1967 [link]
One of the best loved sights of the Roman Arena was a duel to the death between a Gladiator and a Retiarius. The Gladiator, being in armour and carrying a sword, was slow in movement but lethal at close quarters. The Retiarius, having no armour but carrying a net and trident, was most deadly at a distance.
Those who wish to test where the odds lay for themselves will, in these softer days, have to make do with a diagram.
Let us suppose that Gladiator starts at 32 and Retiarius at 1 and that they move in turn. Gladiator moves three circles at each turn and Retiarius four. Both must always move along the lines but can change direction or double back during their move. The duel is won by whoever first lands on top of his opponent at the end of a turn.
Can either player be sure of winning? If so, who?
In the book Tantalizers (1970) a reworded version of this puzzle appears under the title: “The Lion and The Unicorn”.
It is not clear from the puzzle text who moves first, so this code checks for both scenarios.
We can identify the players by the number of steps they take (+3 and +4). I look at games with an increasing number of moves, as defensive play could continue indefinitely, with the players just moving back and forth between two positions.
This Python program runs in 61ms. (Internal runtime is 768µs).
Run: [ @replit ]
Solution: Whichever player takes the first turn can guarantee a win.
If +4 moves first from 1 to 12 (or 14), then +3 can only move from 32 to one of (20, 25, 27, 30), and all of these are +4 moves from 12.
And if +3 moves first from 32 to 25 (or 27) and then to 13, then the squares reachable from 13 in +3 moves are exactly those squares reachable from 1 in +8 moves (and 25 is not reachable from 1 in +4 moves).