The Dishes Puzzle

• Initially, I approached the problem by considering a small group of guests. For a group of 4, I found that there could be 6 dishes of rice (using "4 choose 2"), 4 dishes of broth (using "4 choose 3"), and 1 dish of meat (using "4 choose 4"). Adding them up gives 11 dishes, meaning each group of 4 would have 11 dishes. Since there are 65 dishes in total, I estimated there could be about 6 groups of 4, which suggests there are 24 people. However, this approach falls short because it doesn't account for guests who eat "across groups," meaning the guests can't be neatly divided into groups of 4. This led me to shift toward an algebraic approach.
• I then assumed a fixed number of guests and divided that number by 2, 3, and 4 to represent the number of rice, broth, and meat dishes, respectively. Adding them together should equal 65. Letting "a" represent the number of guests, the equation becomes a/2 + a/3 + a/4 = 65, which gives a = 60.
• Using examples, puzzles, and mathematical histories from diverse cultures can make a significant difference in engagement. This problem was especially engaging for me because I could visualize the setting—a bustling local restaurant with only a few items on the menu. I thought about how the cook might have gathered the information. With this context, I found myself more invested in the math. I also appreciated seeing a math word problem from a non-Western culture, as these are valuable resources that offer unique insights into different ways of thinking mathematically.
• The story and imagery in a word problem or puzzle are crucial. When the imagination is sparked, I explore different ways the problem can be interpreted and solved. The only insight I have is that, if I give students a puzzle with a specific answer in mind, I would let them know upfront to limit their imaginative scope and help them focus on the puzzle’s main point.