# Problem: January Problem to discussion

Hello, all!

We had a lot of fun unpacking the "The X Days of Christmas" discussion on 12/17 and 12/18, so let's continue the fun looking at this Notice and Wonder Problem.  This will be a discussion we start today 1/13 and unpack this week.

Some glasses are black; some are transparent.   I see old chairs... I wonder if that's an old cable hookup on the wall to the right or electrical outlets, and I wonder what is on the wall near the doorway (where heat and/or ac push air out?).   Why is the door open to the outside?

Math:   how long has that food been there?

This reminded me of the banquet table problem. There are 5 chairs on each long side and 1 chair on each short side. This is a total of 12 chairs at the table. What if you had multiple tables lined up short end to short end, jammed up against each other so that you had to remove some chairs? How many people could you accommodate? If two tables were pushed together in a line, if three tables were pushed together in a longer line, etc.

Eric

### Tables and Chairs Photo

I'm first drawn to counting the chairs and observing if there appears to be a place-setting for each one. Looks like 10 chairs on each side and a chair at each end of the table.  So I count 12 places in all.  It did remind me of the banquet tables problem too, as Eric mentioned.  I like the fact that it made me revisit the banquet tables questions, because I find it a good review of linear progressions.  I made a drawing and a values table for 2, 3, and 4 tables if placed end to end, and can observe a pattern.  I find it a good way to find a slope, create a linear equation, and sketch the corresponding graph. Now I suppose a similar exercise could be constructed regarding the plates and chargers pictured, if desired.

But I'm not done exploring; I observe 3 different style chairs.  They too, are set in some kind of pattern:  From the closest viewpoint, they are symmetrically arranged tall ("ladder back"), short, tall, tall, short, and curvy-back chairs at either end.  From the back of the picture, the chairs are arranged short, tall, tall, short, tall, and with the curvy-backs on each end.  This makes me wonder, if a curvy-back were removed and we put another long table at the end, what would the pattern of the chair styles be? (I think it best to stick to just one progression to start, like the front-to back pattern.) It looks like the pattern would continue the same at every added table.  Though now I wonder if there's some fancy equation that could determine an nth setting.  It's out of my league, but thoughts are welcome.

So much math, so much to wonder about here!  How many rungs are there on the chairs in the room?  It appears there are at least three different kinds of chairs.  I wonder about the pattern of drinking glasses that seem to alternate colors.  How many utensils would there be in total?  The most important thing I wonder is when will the food be served?