Measurement Puzzles
A Lesson in the “Math + Fun!” Series
Mar. 2005
Measurement Puzzles
Slide 1
About This Presentation
This presentation is part of the “Math + Fun!” series devised
by Behrooz Parhami, Professor of Computer Engineering at
University of California, Santa Barbara. It was first prepared
for special lessons in mathematics at Goleta Family School
during the 2003-04 and 2004-05 school years. The slides can
be used freely in teaching and in other educational settings.
Unauthorized uses are strictly prohibited. © Behrooz Parhami
Mar. 2005
Edition
Released
First
Mar. 2005
Revised
Measurement Puzzles
Revised
Slide 2
Measurement for Cooking
You have a 3-cup container
and a 5-cup container only.
The cups are not graduated.
How do you measure one
cup of sugar?
1=3+3–5
2=5–3
3
4 = 3 + 3 + 3 – 5 or 1 + 3
5
6=3+3
7 = 5 + 5 – 3 or 2 + 5
8=5+3
9=3+3+3
10 = 5 + 5
Challenge: Continue to 20 cups.
Mar. 2005
5
3
10
Now add a 10-cup container
and show how to measure
11 cups of sugar?
11 = 10 + (3 + 3 – 5)
Measurement Puzzles
Slide 3
Activity 1: Measuring Sugar
You have a 1-cup, a 3-cup, and a 9-cup
container. The cups are not graduated.
How do you measure different amounts
of sugar, from 1 to 30 cups, using the
fewest number of steps? For example,
use 3 once instead of 1 three times.
1
2=1+1
3
4=3+1
5=9–3–1
6=6–3
7=9–3+1
8=9–1
9
10 = 9 + 1
Mar. 2005
11 =
12 =
13 =
14 =
15 =
16 =
17 =
18 =
19 =
20 =
1
3
9
21 =
22 =
23 =
24 =
25 =
Measurement Puzzles
26 =
27 =
28 =
29 =
30 =
Slide 4
How to Get Half a Glass of
Water?
A cylindrical glass is filled to the brim.
How do you get half a glass of its
content without using any other item?
Can’t do this if the glass has a conical,
rounded, or irregular shape.
Full glass
Half full
You have a container, with a tight lid or
cork, filled to the brim with fruit juice.
How do you divide the juice exactly in
half without any measuring tools?
Hint: Suppose the container is exactly
half full. What happens if we close the
lid or cork and turn it upside down?
Mar. 2005
Measurement Puzzles
Slide 5
Measurement at the Hardware Store
A large container is known to hold 24 oz of nails.
We have a balance, but no weights.
Measure out 9 oz of nails for a customer.
Divide all nails into two equal piles
24
12 oz
12 oz
Divide one pile into two equal piles
12 oz
6 oz
6 oz
. . . and again
12 oz
Mar. 2005
Measurement Puzzles
6 oz
3 oz
3 oz
Slide 6
Activity 2: Dividing the Snack Equally
A bag contains carrot coins or
m&m chocolates. You have a
balance, but no weights.
How would you give several
kids an equal share of the
snack? It’s okay to have some
left over, as long as each kid
gets the same amount.
4 kids:
5 kids:
8 kids:
15 kids:
Mar. 2005
Measurement Puzzles
Slide 7
Using only Four Weights
A chemist has a balance and fixed reference
weights of 1, 2, 4, and 8 grams. Show that she
can weigh any amount of material from 1 to 15
grams by putting the fixed weights on one side
and the material to be weighed on the other.
1, 2, 4, and 8 grams are obvious.
3=2+1
5=4+1
6=4+2
7=4+2+1
9=8+1
10 = 8 + 2
11 = 8 + 2 + 1
12 = 8 + 4
13 = 8 + 4 + 1
14 = 8 + 4 + 2
15 = 8 + 4 + 2 + 1
Using 1, 2, 5, 12 gram weights,
Challenge: What are the best four
we can weigh any amount of
weights to use, if we can place
material from 1 to 20 grams.
them on both sides of the balance?
Mar. 2005
Measurement Puzzles
Slide 8
Activity 3: Weighing with Four Weights
Show how to weight any amount from 1 to 40 grams.
1
2=3–1
3
4=3+1
5=9–3–1
6=6–3
7=9–3+1
8=9–1
9
10 = 9 + 1
11 =
12 =
13 =
14 =
15 =
Mar. 2005
16 =
17 =
18 =
19 =
20 =
21 =
22 =
23 =
24 =
25 =
26 =
27 =
28 =
29 =
30 =
1
3
grams
9
grams
27
grams
31 =
32 =
33 =
34 =
35 =
Measurement Puzzles
36 =
37 =
38 =
39 =
40 =
Slide 9
Find the Lighter Counterfeit Coin
We have 3 coins. Two are good coins; one is a counterfeit coin that
weighs less. Identify the counterfeit with one weighing on a balance.
Compare coins 1 & 2.
If they weigh the same, coin 3 is counterfeit;
otherwise the lighter of the two is counterfeit.
We have 9 coins; eight good coins and
a counterfeit coin that weighs less.
Identify the counterfeit with 2 weighings.
Mar. 2005
Measurement Puzzles
Slide 10
Activity 4: Heavier Counterfeit Coin
We have 3 coins. Two are good coins; one is a counterfeit coin that
weighs more. Identify the counterfeit with one weighing on a balance.
We have 9 coins; eight good coins and
a counterfeit coin that weighs more.
Identify the counterfeit with 2 weighings.
Challenge: Find one heavier counterfeit coin
among 27 with 3 weighings on a balance.
Mar. 2005
Measurement Puzzles
Slide 11
Odd Coin: Lighter or Heavier
We have 12 coins. Eleven are good coins; one is a counterfeit that
weighs less or more than a good coin. Identify the counterfeit coin and
its relative weight with a minimum number of weighings on a balance.
Hint: First do it for 3 coins, one of which is a
counterfeit, using only two weighing,
Compare
1&2
If equal, 3 is the counterfeit. Weigh 3 against 1
to see if it is lighter or heavier.
If 1 lighter than 2, coin 3 must be good. Weigh
1 against 3. If equal, 2 is counterfeit & heavier.
Otherwise, 1 is counterfeit & lighter.
If 1 heavier than 2, do as in the previous case.
Mar. 2005
Measurement Puzzles
Slide 12
Odd Coin: Lighter or Heavier (cont.)
We have 12 coins. Eleven are good coins; one is a counterfeit that
weighs less or more than a good coin. Identify the counterfeit coin and
its relative weight with a minimum number of weighings on a balance.
Hint: First divide the coins into three groups of 4 coins: A, B, and C.
Compare
A&B
If A = B, then C contains the counterfeit coin.
Weigh 3 coins from C against 3 good coins.
If equal, the lone remaining coin in C is
counterfeit and one more weighing is enough
to tell if it’s lighter or heavier than a good coin.
Mar. 2005
Measurement Puzzles
If the three C coins
are lighter, then . . .
If the three C coins
are heavier, then . . .
If A < B or A > B . . .
Slide 13
Measurement with Scales
We have 100 oranges in a heavy cardboard box.
How can you find the weight of the oranges
(without the box) using a bathroom scale?
Bathroom scales are usually not accurate for
small weights (say, less than 30 lbs) or very large
weights (say, more than 200 lbs). How would you weigh
a baby who is about 20-25 lbs using a bathroom scale?
How would you weigh the 100 oranges accurately
using the bathroom scale if you estimate the oranges
to weigh 20-30 lbs and the cardboard box 3-5 lbs?
Can you accurately weigh a person who is about
250-300 lbs using two bathroom scales?
Mar. 2005
Measurement Puzzles
Slide 14
Activity 5: Measurement with a Scale
You have 5 packages of candy bars and a scale.
Each candy bar is supposed to weigh 200 grams.
One of the packages, however, contains candy
bars that weigh 150 grams.
Can you find the package with lighter candy bars
using the scale only once?
Extra challenge: You only know that one package
contains lighter candy bars but you don’t know
the weight of regular or lighter bars. How many
times do you need to use the scale to find out the
two weights and the package with lighter bars?
Mar. 2005
Measurement Puzzles
Slide 15
Measuring Length
You are curious about the height of a light
pole in or next to your house. Can you
measure its height without climbing it?
Light
pole
Hint: Think of shadows!
If a 1-foot upright ruler casts
a 6-inch shadow and the
shadow of the light pole is
10 ft, the pole is 20 ft high.
1-foot
ruler
Standing next to a stream that you can’t cross,
you and a friend make a bet. She says that
the stream is 15 ft wide; you say that it is 20 ft
wide. You have a 30-ft piece of string and a
1-ft ruler. How can you decide who is right?
Mar. 2005
Measurement Puzzles
Slide 16
Measuring Volume
How can we measure the volume of an
irregular object such as a chess piece?
1. Put a container filled with water
inside a larger container.
2. Slowly insert the item into the
inner container.
3. Measure the volume of the
water that spills out by pouring
it into measuring cups.
Mar. 2005
Measurement Puzzles
Slide 17
Next Lesson
Thursday, April 14, 2005
1.
2.
3.
4.
5.
Think of a 2-digit number (10-99).
Multiply your number by 5.
Add 500 to the result of step 2.
Add the number of your siblings.
Double the result of step 4.
Mar. 2005
You got a 4-digit number that
starts with 1, has your original
2-digit number in the middle,
and double the number of
your siblings at the end.
Measurement Puzzles
Slide 18