The Dishes Problem

The problem: 

"How many guests are there?" said the official.

"I don't know," said the cook, "but every 2 used a dish of rice, every 3 used a dish of broth, and every 4 used a dish of meat between them". There were 65 dishes in all. How many guests were there?

My solution (without algebra):

1) Assess- guess what the solution might look like. Since the question is asking for number of guests, the value must be positive and must be whole (Natural Numbers).

2) Estimate- every 2 guests used rice, so in 65 dishes, this could be up to 65/2 = 33 guests. Every 3 guests used broth, so this could be up to 65/3 = 22 guests. Every 4 guests used meat, and this could be up to 16 guests. In total, without checking for overlapped dishes, there could roughly 30 + 20 + 15 = 65 guests. The only guests that had rice and broth and meat were the 12th, 24th, 36th, 48th, and 60th, so discounting them from 65 potential guests leaves 60 guests. 

3) Diagram- Notice below that after every 12th guest, a pattern repeats and 13 dishes are complete, so we can set up a ratio.

12 guests = 13 dishes (not to be solved by arithmetic operations)

 x guests    65 dishes

Blue is rice; purple is broth, and pink is meat. 

4) Final answer - *Through trial and error,* we can see that x = 60. This also lines up really well with the estimate in 2) and the number characteristics match our assessment in 1). Hence, there were 60 guests.

Reflections

How could you solve this puzzle without algebra (or at least, without the algebra we are used to)?
Algebra generally refers to the use of symbols and expressions to represent mathematical situations. Despite trying my best to avoid algebra, I simply ended up reactively dodging it. (especially in part 3). So, it is possible to complete without but my instincts were tough to ignore. Trial and error + pictures were solid alternatives! The Egyptian method of False Position would have been a great option too. 

Does it makes a difference to our students to offer examples, puzzles and histories of mathematics from diverse cultures (or from 'their' cultures!)

It does make a difference! When students from various backgrounds are immersed in the euro-western perspective, it helps them to feel confidence and see value in other knowledges by interacting with other methods. It could allow them to understand how math plays a role in other communities and why it's important to understand the techniques and work habits well enough to adapt them. 

Do the word problem or puzzle story and imagery matter? Do they make a difference to our enjoyment in solving it?

Yes, it does. Rather than solving the "dishes problem", we could have easily told students to "solve x/2 + x/3 + x/4 = 65," and that's not half as fun! The more students see mathematics as something that can be approached recreationally, the more interest they will have in it and be able to connect it more broadly. If studentA was the cook in real life, there might have been in trouble if the number of guests were logged incorrectly for the official. However, through just 4 seemingly trivial memories, the answer was revealed. And by using math! Puzzles are wonderful for imagination, problem solving, and collaboration. It encourages more neural pathways and gives that dopamine rush.