A Knight’s Tour: Excelsior by Jaques
The original source of inspiration for this puzzle was Lewis Carroll’s series of mathematical posers known as the Tangled Tale. Though it has to be said that not much of the tale’s first “knot” – Excelsior has made it into the actual crossword.
In Carroll’s puzzle two knights set off to climb up and return from a nearby hill, walking at 4mph on level ground, 3mph when climbing and 6mph in descent. No information is given as to how much flat and how much hill the walk comprises. Walking steadily at these speeds for 6 hours the question is how far do the knights travel?
This is one of those maths questions that exploits human fallibility when considering average speeds, with instinct being so spectacularly wrong that people often fall into the trap even when told in advance that this is going to happen. As the knights descend at twice their speed of ascent they only spend one third of their time on the hill in descent and two thirds climbing. This means their average speed for up and down is 4mph (two thirds of three plus one third of six) the same as on the flat. If, for example, they had 2 miles of climbing and 2 miles of descent it would take 40 minutes going up and 20 minutes down. A total of 1 hour to cover 4 miles.
This means how ever much up and down there is the average speed is 4mph and their overall distance 24 miles in 6 hours.
[Of course, the truly brave puzzle solver can always reason as follows, “I’m not given the amount of climbing in the question, therefore I do not need to know it and the journey’s distance must be independent of terrain. So we will assume the journey is in fact completely flat which would have the knights walking for 6 hours at 4mph and covering 24 miles – QED”]
Having originally thought this would make an interesting puzzle this was almost immediately followed with the second thought “er, no it wouldn’t”, as there seemed no way of including any reasonable amount of the original puzzle into a crossword’s set-up. The question was too fiddly for the solver to discover during solving and baldly stating the thing up front seemed to bypass the crossword connection altogether.
However, the idea of knight’s movement had conjured up the old crossword favourite of a knight’s tour. Coupling this with seeing A. B. Frost’s illustration of 2 knights descending the hill in perfect lock-step raised the question is it possible to produce 2 simultaneous knight’s tours of a chessboard which have the knights always landing in adjacent squares?
Out with the chessboard, sixty four small numbered squares of paper and a large stack of pennies to help keep track.
In the end devising the routes was not as difficult as had been anticipated. Each 4×4 quarter of the board has 4 little sub-routes that the knights follow, either a loop around the quarter’s edge or a dive through its centre in and out of 2 opposite corners. If one of the knights is doing one of these while the other is doing the other they remain side by side throughout that short section
This split the routes into 16 short sections for each knight and not too much juggling about was necessary to join these sub-routes together for both knights to make a coherent whole. Only one awkward jump diagonally across the centre of the board and one occasion remaining within the same quarter was needed to knit everything together.
With routes entirely mapped it was possible to manufacture a chart showing how letters on the board related to letters in each knight’s tour. So putting the final message on the board established a dozen letters in each tour and then generally alternating between the knights made it possible to select a word for one and examine the consequences this had for the other. The answer GRAIL was included purely to amuse the setter.
Once again filling the grid turned out easier than expected even with the added requirements of the two knights having the same number of words and there being at least one pair of adjacent squares where both knights began a word in order to give a start to the routes. In fact, insisting that the knights never started a word in the same square (which might have created a difficulty in numbering) still didn’t bring the edifice crashing down. Twelve words for each knight gave an average word length of under 5.5, so perhaps eleven or only ten words would have been better. But, it was hoped clues with short answers would be easier solve and be of benefit to the solver.
The big question remaining was “Can this actually be solved?” Obviously, solving all 24 clues cold would provide the letters and the routes can undoubtedly be deduced via a logical path from theses answers, but still leaves the problem of fairness.
There are plenty of small hints the solver can guess at involving such things as alternation of black and white cells (the main reason why the grid was unnecessarily coloured like a chessboard); forced moves joining 3 cells together through each corner; occasionally limited routes from one number to the next and the need for the knights to remain in adjacent squares throughout. Even spotting the appearing final message might help. So it was hoped the solver would not have to cold solve an excessive number of clues to reach the end.