Word problem

Question

Bearsandme, Peas & Corn, At play by jc/eaglei, and Cascadeclimber sat down in the room to paint bird houses for the Audubon Bird House Event Cache. Their instructions were that each time they finished painting a bird house they would carry it to walkingron to be placed on a shelf with the other bird houses also being painted by others in the room, and then return to paint another house until all their houses were done. They all worked hard at painting their houses and the approaches varied from relatively simple to extremely elaborate. They concentrated on their own work for the most part, but from time to time while chatting each looked up to see the others painting, but no one paid much attention, so none of them knew how many houses any of the others had finished. After a while cascadeclimber, after giving walkingron a completed bird house, started to reach for another unpainted house, but as he did he saw, as they all saw, there was only one house remaining . The rest had all been painted. Who should paint the last bird house? At that moment walkingron came back into the room and saw their confusion. "It's very simple," he said. "Whoever has painted the fewest houses should paint the last one. If there's a tie, you share it. I gave the four of you a dozen houses to start with and I know I've seen you all come up at least once with a painted house in your hand. Do you all know how many houses you painted?" They all answered “yes.” So walkingron continued, "Bearsandme, did you paint more houses than Peas & Corn?" She replied, "I don't know." "Cascadeclimber, did you paint more houses than Bearsandme?" He replied, "I don't know." Then from out of nowhere a voice from the back of the room blurted out, “Aha” said At play by jc/eaglei, “We know who should paint the last house. In fact, we know exactly how many houses each of the four of us had painted.” ….. Do you? You must assume that each of them answered truthfully and that each of them, being a superb logician, made all possible conclusions that could be made with certainty from the available information at the time they gave their answers. Let "B”, "P”, “A”, and "C” represent the number of bird houses painted by Bearsandme, Peas & Corn, At play by jc/eaglei, and Cascadeclimber respectively

Answer 1

When C gave Walkingron his completed house, there were 10 houses already painted on the shelf, so, including the one C painted, 11 houses had been painted, because there were 12 unpainted houses originally.

Each painter knew how many houses he/she had painted but didn’t know how many the others had painted, nor could they compare one another’s efforts. Walkingron said that each painter had completed painting at least one house.

A knew how many houses he/she had painted.

In order, the number of houses painted by the four painters could be:

{1 1 1 8}, {1 1 2 7}, {1 1 3 6}, {1 1 4 5},  {1 2 2 6},

{1 2 3 5}, {1 2 4 4}, {1 3 3 4}, {2 3 3 3}.

A knew how many houses he/she had painted. If it was 8, then the other three would only have painted one each. If it was any other number, A could not have deduced how many the others had painted.

So we can say A painted 8, while B, C and P painted only one each. B, C and P would share in the painting of the last house.