I tried this problem (from Thinking Mathematically by Mason, Burton and Stacey) today, posted it on Twitter, and also gave it to some friends.
I was particularly interested in how different people tackled the problem with regards to creating representations and notation. For me, the question is then:
What does all this tell us about what people do when they do mathematics?
Here is Martha's attempt. She created one of those tables that you get for 'logic puzzles', but found it difficult to incorporate the fact that each person had two jobs:
Next is Harry's attempt. He found it hard to explain what he had done exactly, but said that he "went by the names only", and at some point had two options and chose one of them to see if it worked out:
This idea of choosing an option and seeing what happened was also employed by @WoodhouseCol (sorry, I don't know your name!):
The tweet above came in response to this tweet from Mark, who tried to put the information into a table but abandoned it in favour of a different approach:
This resonated with my experience. I considered using a table with 'job 1' and 'job 2', but was not sure how or if it would work, so abandoned it (more on this below).
Unfortunately I don't have images of Mark's approach, but it sounds similar to Sarah's, who started with a table of the jobs that did not go together - see the top left post-it in the image below - then shifting to using only the words for names and jobs whilst referring to this table.
She did it all on two post-it notes (top left and bottom right!), but I asked her to set out each stage of her thinking on a separate note to show the steps between:
Douglas also used only names, but used a bold / strike-through scheme to help clarify what was possible and not possible (click to view):
Sam switched between two tables, one of which was similar to Sarah's first table showing which jobs could not go together (click to view):
Suzanne created a number of images in order to help her 'enter into the problem', as can be seen below (click to view); notice how she too stressed the jobs rather than the names:
Her husband represented the information in a network:
Stuart also represented the information in a network, with the addition of colour (click to view):
Simon also used a network (click to view):
I used a network too, after an initial sketchy attempt and realising the problem was as much about what couldn't go together as what could. I used straight line for a match, and wavy lines for a non-match. The little wavy lines connecting jobs that didn't go together - on the very right-hand side of the images - were the key to unlocking the problem for me. The final image is a redrawing of the middle image in order to simplify and make further connections.
Finally, here are a series of tweets from David, who has been exploring what happens when showing his mathematical vulnerability on twitter recently:
I am fascinated by how this puzzle leads people to unexpectedly adjust (or abandon) representations and notation during the process, as can be seen in this work in progress from Dan: