Starting from the idea of the same clues and same grid producing different solutions in different bases, the initial question was - how to convey to the solver what is actually required? This needed to done during the course of the puzzle in order to justify the initial base 10 solve.

Weaving messages into numericals is not as straight-forward as for those found in word puzzles and the best approach seemed to be to include the instructions in the grid. This had the advantage of being able to be explicitly flagged in the preamble so solvers knew what to look for.

The biggest drawback of this method was the second grid would have only 9 different digits (0-8) to chose from, meaning the encoded message had to be written using just 9 different letters. So the first major problem was producing a relatively clear and concise message that would fit neatly into a grid using just 9 letters of the alphabet. Considerable amounts of hair and white-board marker were wasted at this stage.

With the message selected the grid dimensions had also effectively been defined. It seemed best to place the words in the central two rows for symmetry and gathering all the letters together would allow for as few lights as possible passing through these critical cells. At this stage the only decision about letter-values that had been reached was which letters would fall in the range 0-8. Which additional letters would be used and even how many variables were kept back as extra “degrees of freedom” for later on.

It was at this point that the true secret of the puzzle was deployed - designing the grid by placement of the bars. The grid is very close to being what is known as “simply connected” that is to say all the bars (including the outside edge) are joined. Only the island of bars in the centre stand in isolation. The even dimensions of the grid and the desire for eye-pleasing symmetry prevented an entirely simply-connected effort. The upshot of producing a grid like this means there are very few alternative routes from one light to another. This would be quite tedious in a word puzzle, where letters entered in the grid are the general way of gaining feedback and the solver tends to slowly build to the solution. Whereas the more peripatetic nature of numericals, where a deduction can produce information all over the place, allows for a much more constricted grid design including higher percentages of unches.

There were now 6 across lights entirely defined in both grids by the letters of the message and a certain amount of playing around with the first 9 letter-values was done to produce relatively “nice” entry values. Generally speaking smaller numbers made up of lots of factors are easy to clue than walloping great primes. The need to avoid leading zeros left L and S as the only possibly candidates to be zero, one in each base, so that was a starting point. A certain amount of back-tracking and moving of bars was done in an attempt to improve the overall set of target values for entry.

It was then necessary, with the aid of some computer power, to consider these 6 acrosses along with any down lights where the majority of the cells were already fixed en masse. All the remaining flexibility as to how many variables and which double-digit value was which was used up at this stage finding suitable clues for these entirely and mostly predefined answers.

Now the simply-connected aspect came into its own. With just about all values assigned in both bases, the grids had their central rows full and a few digits sticking out into the top and bottom sections. By considering each light in turn working out from the middle it was possible find a double (base 9 and base 10) clue where only 1 or perhaps 2 of the light’s digits were fixed. Each new light generally entering digits into virgin territory, meaning there was never a point where an entry was fully decided upon before its clue had been sought.

There was, however, a price to be paid for the nature of the grid. The lack of inter-connectedness and the high number of unches meant not enough information could be gleaned in grid feedback. In short the puzzle was completely unsolvable. It was possible to return to an earlier stage and try again, but this merely produced a different, equally unsolvable puzzle.

A desperate remedy was required. Hence the introduction of equals clues. Not the most elegant solution to the problem, but when the lifeboat is sinking something has to be chucked over the side! Fortunately the manner of considering each light in turn was good enough to allow some clues to have alternatives that produced the same entry values in both bases. Test solving allowed a suitable number of these to be included to leave a reasonable route to the solution.

The only remaining question was - is it fair to ask solvers to deduce that 2 digits must be entered in the odd, earmarked space? Essentially this is the only logical solution and solvers must abandon all prejudice and read “All entries are in base 10₉” correctly as “All entries are in base NINE”.

Posted on Monday, January 25th, 2010 at 10:44 am under Setting the scene.
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