Download © Jan Gombert, 2006 The Chicken Scale Problem Problem Set 1

The Chicken Scale Problem
You own an old-fashioned grocery store, and have a balance scale for weighing
chickens. Assume that all of the chickens you sell are a whole number of pounds. You
want to select a set of weights for measuring these chickens.
In all of the problems below, you want to be able to weigh
(a) the heaviest chicken possible, and
(b) all chickens that are less than the maximum, down to one pound.
For example, if you choose two 1 pound weights, (a) the heaviest chicken
you could weigh would be 2 pounds, and (b) you could also weigh a 1 pound chicken. If you choose a 1
pound weight and a 5 pound weight, you could weigh a 1 pound chicken, a 5 pound chicken, and a 6
pound chicken, but you would not be able to weigh a 2, 3 or 4 pound chicken.
Go back and read the sentence in the box above again.
Problem Set 1: Chicken on the right side, weights on the left side.
Problem 1. With two weights, what is the heaviest chicken you can weigh? (Given that you must be
able to weigh all lighter chickens, as well.) What are those two weights? Make a table to check your
answer. For example, suppose that you are using two 1 pounds weights. Your table would look like
this:
weights
1
1, 1
chicken weight
1
2
And the heaviest chicken you could weigh would be 2 pounds.
Problem 2. With three weights, what is the heaviest chicken you can weigh? What are those three
weights?
Problem 3. Solve the problem for four weights.
Problem 4. Take four weights: 1, 2, 3 and 4 pounds. (This is not the answer to problem 3.) (a) What is
the heaviest chicken you can weigh with these? (b) Can you measure everything from this down to one
pound? (c) Make a table of all possible combinations of these four weights, together with the total
weight for each combination. Here, for example, is how the table might start.
weights
1
1, 2
1, 3
chicken weight
1
3
4
(d) Are there any duplicates in these total weights?
07-TheScaleProblem.doc
© Jan Gombert, 2006
(e) Now make a similar table using your weights for your answer to problem 3. (f) Are there any
duplicates in those weights? Could you conclude that your answer to problem 3 is the most efficient?
That is, you get the most from the least number of weights.
Problem 5. Think of the four weights (your answer to problem 3) sitting in front of you on a table. With
each weight, you have 2 choices: you can put it on the scale or you can leave it on the table. This is like
flipping four pennies. If a penny comes up heads you put the weight on the scale, otherwise you leave
it on the table. (a) How many different ways can four pennies come up? (HHHH, HHHT, etc.) (b) Does
this agree with the number of combinations of weights you got in problem 4? If not, why not? What
would account for any difference?
Problem 6. Now generalize problem 3 to n weights. (a) What is the heaviest chicken you can weigh
with n weights? (b) What is the pattern of the weights you have chosen?
Problem Set 2: Chicken on the right side, weights on either side.
In this set of problems, you can put some of the weights on the side of the scale where the chicken is.
For example, if you have a 1 pound weight and a 5 pound weight, you can weigh a 4 pound chicken.
Put the 1 pound weight on the right side (with the chicken) and the 5 pound weight on the left side.
Both sides of the scale have 5 pounds.
Problem 7. What is the heaviest chicken you can weigh with two weights? Again, you need to be able
to weigh that plus all lighter chickens.
Problem 8. What is the heaviest chicken you can weigh with three weights? (a) When you come up
with an answer, make a table listing all combinations of those three weights. Here is the start of such a
table, using the weights 1, 5, and 6 pounds.
weight on left
1
5
1, 5
6
weight on right
(with chicken)
1
weight of chicken
1
4
6
6
Make sure that you listed all possible combinations of the three weights, without regard to the chicken
weight. For example, in the table above, all of the weight combinations on the left and right are
different, but the chicken weight of 6 pounds is duplicated. (b) In your table, are any of the weights
duplicated? This might suggest to you (as in problem 4) that there is a better answer.
Problem 9. Solve the problem with four weights. Use the test in problem 8 (for duplicates) to verify that
you have the best answer.
Problem 10. Generalize the problem to n weights. (a) What is the heaviest chicken you can weigh with
n weights? (b) What is the pattern of the weights you have chosen?
07-TheScaleProblem.doc
© Jan Gombert, 2006
Teaching Notes
Problem set 1 is really about binary numbers. With each weight, you can either put it on the scale or
not, leading to 2n possible weight combinations. The optimal solution is then to have weights of 1, 2, 4,
etc. points.
Problem set 2 is about base 3 numbers.
I’ve yet to take a class to this level of understanding.
07-TheScaleProblem.doc
© Jan Gombert, 2006