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.

Reflection - Solving ancient puzzles in ancient and modern ways

TsáKtalay’pa and I tackled the proof of the formula for volume of a pyramid: V=1/3*base area*height, in ancient and modern ways. The original problem suggests investigating through two special cases, dividing a cube into six congruent symmetrical pyramids, and dividing it into three congruent oblique pyramids. We argued that the Ancient Egyptians likely derived the formula not from geometric dissections of a cube but through practical building methods. They likely began with a frustum for structural stability and efficiency of materials, and continued building upward to form a complete pyramid, perhaps aiming to connect with the stars. For the modern proof, the intuitions from these cases were helpful. We provided clay models for our classmates to explore the two divisions of the cube, along with paper templates to visualize the second case of three asymmetrical pyramids. We extended the discussion by introducing the Mayan pyramid, El Castillo, which resembles stacked square prisms and provided intuition for the formal proof using integration. This comparative approach enriched our understanding of the formula, blending practical, historical, and mathematical perspectives.

I learned a lot from both the process and content of this presentation. It sparked my curiosity about formulas I had previously taken for granted. Taking the time to explore and question them helped strengthen my relational understanding and appreciation of math, making me feel more prepared as a future math teacher. Understanding these concepts through ancient methods added historical context and narrative to the formulas, enriching the learning experience. I especially enjoyed the "extension" part of the presentation—not only was the research helpful, but it also encouraged me to connect and expand on different areas of my knowledge.

It was also really inspiring to see my classmates' presentations. I learned a great deal from this project, and I plan to adapt it into a class activity in the future.