Elementary POW, March 4-8, 1996
Elementary POW Problems || January-March, 1996 Problems || Elementary POW Main Page
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Elementary Problem of the Week, March 4-8,1996
******* ******* *******
* PP * * NN * * NP *
* * * * * *
******* ******* *******
There are 3 boxes. One contains 2 pennies, another contains 2 nickels, and
the third contains 1 penny and a nickel. Each of the boxes is labeled PP,
NN, or NP; but each is labeled incorrectly.
If you were allowed to take one coin at a time out of any box, what is the
smallest number of drawings needed to determine the contents of each box?
Explain your answer.
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Bonus Puzzler
The product of a number times itself is called a "square" number. For
example, 1, 4, and 9 are all "square" numbers since 1=1x1, 4=2x2, and
9=3x3.
The word square is borrowed from geometry. Why do you think these numbers
are called "square" numbers?
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Thank you to everyone who replied to the POW. There were many correct
solutions. Special congratulations to Alexandra Reekie, Joshua Branfman,
Conrad Kirby, and Carolyn Rosenthal for getting an answer of 1.
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Alexandra Reekie
Grade 5
Western Salisbury Elementary
Allentown, PA.
Teacher: Mrs. Geschel
You need one drawing.
Since each box is labeled incorrectly, draw one coin from the NP box. If it
is a penny (nickel), then it has to be the box containing two pennies
(nickels). Then you know that the NN (PP) box cannot contain two nickels
(pennies), so it must contain one nickel and one penny, and the PP (NN) box
must contain two nickels (pennies).
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Joshua Branfman
Grade 3
Teacher: Paul Nass
Georgetown Day School
I think the answer is one because if you picked from the box that was
labeled NP and you got a penny you would know that that would not be the NP
box because the box was labeled NP and all the boxes were labeled wrong.
And you would know that it wouldn't be the NN box because you picked a
penny.
This means that the box labeled NN must have a penny and a nickel because
it couldn't have the two pennies because the box labeled NP had the two
pennies and it couldn't have two nickels because all the boxes were labeled
wrong.
This leaves the box labeled PP having the two nickels.
If you pulled a nickel out of the box labeled NP it would all happen
backwards.
Bonus Puzzler
The area of the square equals to the width times the length. In a square
the width equals the length. So the area of a square is always a number
times itself.
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Conrad Kirby
4th grade
Burton Elementary School
Durham, NC
Question: If you were to take 1 coin at a time out of any box, what is
the smallest number of drawings needed to determine the contets of
each box if all of the boxes are labled incorrectly?
****** ****** ******
* NP * * PP * * NN *
* * * * * *
****** ****** ******
Answer: 1 Drawing (but it has to be from box "NP".)
Explanation: You pick out a penny from box "NP" and you know it is has
to be PP. This is because the boxes are mislabeled: it can only be PP
or NN. Then you know the other boxes could be NN or NP. But as you say
all of the boxes are mislabeled. That leaves you with box "NN" being
NP and "PP" being NN.
It works the same if you take a nickel out of box "NP": "NP"=NN;
therefore "NN"=PP and "PP"=NP.
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Carolyn Rosenthal
Paul Nass
3rd grade
Georgetown Day School.
You have a cup labeled P.N. you pick a penny out, so it has to be P.P.,
P.P. has to be N.N., and N.N. has to be P.N. that is how you do it in one
turn!!!
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JOHN, PATRICK, GIULIANA, LIZZIE, VICKIE, CHRIS, TED, JESSICA, DAN,
BARBARA, SIMON, KERI, AND STEPHEN.
4th Grade
Drexel Hill School of the Holy Child
Mrs. Yantosh's class.
We think the smallest number needed is three. The answer to the problem
is: you start with box NP. You pick one coin at a time and then you pick up
one coin out of any of the two boxes. Whatever comes out of the box the
duplicate is in the same box. The third box will have the other PP or NN
combonation.
Our bonus answer;
The word square is used for these numbers,1,4,9 because if we used a dot to
represent each number {4=....} {9=.........} you can form a square. You
can not form a square with numbers like 2,3,4,7
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Dan, Matt and Greg. Munsey Park Problem Solvers from Miss Duggan's Class
We are responding to your bonus puzzler to the problem we sent in a few days
ago. Our answer is that if you square a number you can draw a square out of
that many objects. EX: xxx
xxx
xxx
This makes the square number nine.
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Kim Fugok, Christine McGowan, Amanda Tumminelli, and Ryan Grace
Grade 6
Drexel Hill School of the Holy Child
Drexel Hill, PA
Caroline Brennan
BONUS
Each number can be arranged into a perfect square. We drew each one on the
board to prove it.
Example: 9 = 3 x 3
__ __ __
|__|__|__|
|__|__|__|
|__|__|__|
The number nine would look like the figure above, a square of side 3.
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AneilB@aol.com
If the first two drawings are from the mixed box, then you can do the
problem in 3 drawings. The first two drawings are 1penny and 1 nickel. If
you draw from the next box and if you get a penny you know that that box
has 2 pennies and the one you have not touched has 2 nickels.
If your first two drawings are from the 2 pennies or the 2 nickels boxes,
then you can do the problem in 3 or 4 drawings. The first two drawings are
2 pennies or 2 nickels. If you take the 2 pennies out of the first box and
move to another and pull out a penny then you know that the box has 1 penny
and 1 nickel and that the third box has 2 nickels. If you did the same
thing but instead of pulling out a penny from the 2nd box you pull out a
nickel, then you don't know which is the 2 nickels and the 1 penny and the
1 nickel. You need a fourth draw to solve this one. If you take two nickels
out of the first box and move on to another and pull out a nickel then you
know that the box has 1 penny and 1 nickel and that the third box has 2
pennies. If you did the same thing but instead of pulling out a nickel you
pull out a penny you need a fourth drawing to solve this problem.
Bonus
There are called square because in geometry a square is a 4 equal sided
figure. If one side is 3in long the other sides are 3in also. To find how
many square inches of area in a square there are you times one side times
another ( 3x3= 9. 9 is a square number.)
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Cecilia Ekperi
3rd grade
Paul Nass
Georgetown Day School
When you pick a coin from each bag, you would have made 3 picks.
The combination for these picks, would be either 2 Ps and 1 N or 2 Ns and 1
P.
If 2 Ps and 1 N is picked, then there would be 2 Ns and 1 P left in the bags.
If 2 Ns and a P is picked, then there would be 2 Ps and an N left in the
bags.
Therefore 1 more pick from any of the bags, for a total of 4 picks, would
give at least1 combination: that is either PP, NN, or NP.
When this happens, the remaining contents of the other bags would then be
determined.
The answer is 4 picks
Bonus Puzzler
In geometry, a square is a 4 sided figure, with all the sides equal.
The area of a square, is the product of 2 of the sides.
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Anna Margush
Grade 4
Home School
Akron, Ohio
Pick two coins from one box:
1.If there is a nickel and a penny
***** ***** *****
* N * * * * *
* P * * * * *
***** ***** *****
Then pick one more coin. If it is a penny, then that box has 2 pennies,
the other has two nickels. If it is a nickel, then that box has 2
nickels, the other two pennies.
1.If the coins are the same, pick one more. If it is the same, then all
the other coins are nickels, so we know the box names.
***** ***** *****
* P * * P * * *
* P * * * * *
***** ***** *****
Unfortunately, if it is different from the other coins, we must pick a
fourth coin from the same box.
***** ***** *****
* P * * N * * *
* P * * ? * * *
***** ***** *****
This will allow all the boxes to be labeled.
We could start by picking 2 coins from 2 different boxes. If they are
the same, we know the last box is PP (or NN).
***** ***** *****
* N * * N * * *
* * * * * *
***** ***** *****
We still have to pick one more coin from one of the first two boxes to
identify them.
If the two coins are different,
***** ***** *****
* N * * P * * *
* * * * * *
***** ***** *****
another coin should be picked from one of the first 2 boxes
***** ***** *****
* N * * P * * *
* ? * * * * *
***** ***** *****
If it is different from the other coin, all boxes can be labeled, but if
it is the same, a 4th coin will need to be picked.
Conclusion: at least 3 coins must be picked, sometimes a 4th will be
needed.
I think that it is called a square because a square has the same thing on
each side.
9=3*3 and a square has 3*3 to get the area which is 9.
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Brittany Stanford
Grade 3
Mrs. Alderete
Frank West Elemetary
Bakersfield, CA
We think the minimum number of drawings of coins you would need to see to
figure out what was in each box is 3 if these were drawings of the same
coins;One from one box and two from another. In other words, if coins were
picked out of the boxes, you could figure it out if the coins picked ended up
being all the same. For example, if three pennies were drawn it would be
obvious that one of the boxes had a PP combitation. The other would have to
have a NP and the one left with no coins revealed or drawn would be the NN .
One from one box and two from another. Otherwise you need to see four coins
if
three of the same are not shown.
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Chaz Rosenberg
Mrs. Pensa
3rd Grade
Center School
Stow MA
The least number of drawings are 3 drawings. You draw 2 coins from 1
box and 1 coin from the next box. Say you draw a nickel and a penny from
1 box and a penny from the next box. You now know that box 1 is a nickel
and a penny, box 2 is 2 pennys, and box 3 is 2 nickels.
BONUS
1, 4, and 9 are called square numbers becau1 by 1, 2 by 2, and 3 by 3s
are all squares!!
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