Math Puzzle: Dishes Problem

Solving by algebra

Define variables and gather information given:

• Let number of guests = g
• Dishes of rice 
• R = x/2 (take the floor)
• Dishes of broth
• B = x/3 (take the floor)
• Dishes of meat 
• M = x/4 (take the floor)
• Total number of dishes 
• T = 65

T = R + B + T

65 = (x/2) + (x/3) + (x/4)

65 = (6x/12) + (4x/12) + (3x/12)

65 = 13x/12

780 = 13x

60 =  x

Since x is a whole number, we do not need to worry about taking the floor of R, B, M, and their sum.

Check: 

• R = 60/2 = 30
• B = 60/3 = 20
• M = 60/4 = 15
• T = R + B + M = 30 + 20 + 15 = 65
• Therefore, there are 60 guests if there are 65 dishes.

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Reflection

How could you solve this puzzle without algebra (or at least, without the algebra we are used to)?

    Without algebra, I would use a trial and error system.  I would see how many dishes there would be if there were 12 people, for example, since 2, 3, 4 can all go into 12 equal.  

If there are 12 people, there would be 6 dishes of rice, 4 dishes of broth, and 3 dishes of meat.  Therefore, there would be 13 dishes for 12 people.  

    I observe that the number of dishes is a number slightly bigger than the number of people.  I can use the number of dishes, 13, and double the number of people until I get close to 60 dishes:

If there are 24 people, there will be 26 dishes. 

If there are 48 people, there will be 52 dishes.  

13 dishes are missing to get to 60, and we know that if there are 12 people, there will be 13 dishes.

So we can add that to the 52 dishes for 48 people above.
So for 60 dishes (13 + 52), there will be 60 people (48 + 12). 

 

    This method is similar to how Babylonians would "multiply" numbers by doubling numbers and repeated addition. 

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

    Yes, I believe it does.  Similarly to movies and other media, when there is representation of one's own culture, especially a minority group, it an evoke a connection to the problem.  For this dishes problem that came from China, I paint a picture in my mind that is reflective of my experiences.  I imagine a big seafood restaurant where a banquet is going on.  The chefs in the kitchen are preparing multiple dishes at a time and ringing them up nonstop.  It's almost silly to think that a math problem can make me reminisce about my past travels to China, but I think it makes the problem more meaningful to me.

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

    I find word problems accompanied with a story and imagery enjoyable.  I can see it in context, whether it is realistic or not.  I can imagine the world through mathematics, and I find it easier to think about.  I believe it is similar to students learning about fractions by looking at a pie.  Another example is using apples in place of just numbers to teach addition or subtraction (I have 2 apples, Jin has 5 apples.  How many apples are there in total?).  There is movement in the problem which can make it easy to imagine when I can use physical objects to visual the problem.