King Arthur was slumped on his throne, glaring at his squabbling knights. Time to fire them up a bit, he thought.
"Right! All of you sit down! Time for another Grail quest."
The knights groaned; misery soon turned into panic. Mathematical ability had never been high on the list of skills necessary for knighthood. Merlin was the smart one and nobody was in the least bit capable of challenging him on his home turf. However, Sir Rod the Obtuse approached Merlin; they exchanged a few words and the knight then slithered off with a faint smirk.
The Round Table was designed so that the chairs were numbered in sequence 0, 1, 2, 3, ... n, where n was the number of knights present. The chair numbered 0 was always empty, ready for the return of a triumphant knight who would finally find the Grail. King Arthur sat on an unnumbered throne.
Once all the knights were seated, Arthur would take out some dice from a box and roll them to get a number, k. Arthur would then count around the Table, starting at 0, and eliminate the knight seated in every k-th chair. He would continue in this fashion until only one chair was left. If the last remaining chair was the 0-chair, then nobody would set out to find the Grail - but King Arthur still enjoyed the sport of making his knights sweat!
For example, if Arthur rolls a 4, and there are only 5 knights present, then the chairs would be numbered (0, 1, 2, 3, 4, 5) and sequence of eliminated chairs would be (3, 1, 0, 2, 5). The knight sitting in chair number 4 would be packing his bags and saddling his horse.
On this particular day, there were 12 knights present and King Arthur rolled two standard six-sided dice. Sir Rod the Obtuse had no intention of riding off to his doom, nor of returning to find himself dispossessed. Which numbered chair should he sit in to avoid being chosen for the Grail quest?
adapted from my own Gifted Mathematics blog.
image from Wikimedia
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