From: LIMBERJ@mail.firn.edu
Edna Evans
Grade 9
School: Martin County High School, Stuart, FL
After many hours of arranging these little shapes around until I
had memorized each one and its defects, I came to the conclusion
that only those that are divisible by four were able to be put
into a square. That means that two tangrams would form a square
and so would four and eight, twelve, sixteen etc.
I came to this conclusion after trying to put all of them
together. I started with two sets of tangrams and worked with them
until they fit. Next, I did it with three of them (which didn't
work). I had figured out beforehand that four would fit because if
you put four squares together, you get a bigger square. Anyway, I
continued until I found that they weren't working any more.
My theory to back this evidence up is that if you have a square,
the first thing that you must remember is that any multiple of
four will make another one. Next you must realize that an odd
number will not do it because squares are even objects (four
sides). This provides the reason for those certain numbers to work
and other ones not to.
By the way, you need to get something that takes a little less
time to do, because I spent half of my Sunday doing this thing!
From: LIMBERJ@mail.firn.edu
Dustin Hicks
Grade 9
School: Martin County High School, Stuart, FL
To start off the problem, I found the area of each piece of the
tangram given as a model. However, since you stated that we were
to use all the pieces I figured that each separate piece's area
didn't matter. I used the area of 64 units squared (that of the
model) and multiplied it by different numbers which represents the
number of sets. I noted that since all the areas of the smaller
pieces were perfect numbers that the total areas were to be
perfect numbers as well. Also, since I was looking for squares, I
needed to find perfect squares as numbers. So, in conclusion, I
found a square number of sets must be used in order for the
tangram pieces to form a square. Please remember that I was going
along with the fact that all pieces must be used.
From: LIMBERJ@mail.firn.edu
Doug Robeson
Grade 9
School: Martin County High School, Stuart, FL
It works with one set and it works with two sets if you put the
two big triangles that you made to be the corners of the square
together and you do the same with the other side.
__________
this corner-\ / l
l \ /l l
l \ / l /l
l / \l/ l
l_/__\_/___l-this corner
___________________
l\ l / l
l \ l / l
l \ l /_____l
l \ l / l l
l________\/ l_____l
l /\ l\ l
l / \ l \ l
l /________\ \l
l / /l l \ l
l/_____/__l___l___\_l
It will make a square every time that you double the previous
number of sets that you have. For example, 1, 2, 4, 8, 16, 32, 64,
128... and so on. It will work when you double it because you will
have enough sets to make another big triangle to have a corner.
From: LIMBERJ@mail.firn.edu
Sara Holtzman
School: Martin County High School, Stuart, FL
In this problem, all perfect squares work and all perfect*2.
I also figured out that two squares work.
------------
| \ | / |
|___\_|/___|
| /\ |
| / | \ |
------------
This picture is something like the one I drew for two.
_________
| | |
|---|---|
| | |
---------
From: LIMBERJ@mail.firn.edu
Christen Bogenrief
School: Martin County High School, Stuart, FL
The answer for the problem of the month is that any number (of
tangrams) that can be square rooted will work, or in other words
if it's a perfect square.
An example of this is that 2 will not work but 4 will; 25 will
work but 30 will not. This is so because you are making a square
and the number (of tangrams) length of the side must be equal to
the width.
From: LIMBERJ@mail.firn.edu
Julia Schumm
School: Martin County High School, Stuart, FL
A square can be made from one set of tangrams, also from two, but
not from three. Four sets also make a square. This leads me to
believe that numbers of sets divisible only by one, two, and four
will make a perfect square of these sets of tangrams.
I proved this by trying with five, six, seven, etc. sets. It did
not work again until eight.
From: LIMBERJ@mail.firn.edu
Danielle P. Cegelis
Grade 12
School: Martin County High School, Stuart, FL
It is safe to assume that for all numbers that are perfect squares
(4,9,16,25,etc...) as values for N, the pieces of the tangrams
will make a larger square. For example if N had the value of 4,
then the new square would be 2 tangrams by 2 tangrams in size.
From: LIMBERJ@mail.firn.edu
Laura Ejups
Grade 10
School: Martin County High School, Stuart, FL
It is possible to make a square with 1, 2, 4, 8, 12 tangram sets,
and so on. Numbers like 3, 5, 7, 10 and 11 will not work.
I found this out by making about 8 copies of the tangrams. Then I
cut each out and started fitting them together. One reason that I
found to explain the sets that work is that they are rep-tiles
(like those that I learned from a previous problem of the week).
That is, the working sets can each be divided into congruent
squares, (each set can be cut into 2 large congruent triangles).
Another reason for a few of the working sets is that some numbers
can be squared (2sq=4, 3sq=9, 4sq-16). The odd (non-working) sets
will not work because they do not fit together properly. They have
an even number of inner triangles (inside the squares) that simply
do not fit together.
From: LIMBERJ@mail.firn.edu
Eric Petersen
School: Martin County High School, Stuart, FL
First, I found the areas of each piece of the tangram. I used the
area of 64 units squared and multiplied it by the different
numbers given in the problem. Since all the areas of the smaller
pieces were perfect numbers, all the areas of the big pieces had to
be perfect too. Since I was looking for square shapes, I needed a
square number. I found that the square number sets must be used
in order for the tangram pieces to form a square.
From: "Marc C. Johnson"
Angie Clark and Joanna Olson
10th grade
Hinckley-Finlayson High School
We have come to the conclusion that for n sets, n needs to equal a
perfect square for all the pieces to fit together in a perfect
square.
We cut out many sets, and tried mixing the pieces from 2, 3, and 4
sets. The only one that worked was the set of 4. It worked easily
because it made 4 individual squares (like the template). They fit
together to make a square. We then conlcluded that sets like 9,
16, 25, etc. would also work because you could make individual
squares like the one with 4 sets. The other sets didn't work -
they made rectangles. They always had an odd length that we
couldn't get rid of.
From: ssafavi@vaultbbs.com
Sean Mostafavi
Grade: 10
School: Smoky Hill High School
You can make such a square if the number N is either
a perfect (numerical) square or twice a perfect square.
So N could be 1, 2, 4, 8, 9, 16, 18, 25, 32, ...
but not 3, 5, 6, 7, 10, ...
The reason? Well, here is one way to look at it:
Take the length of the square made from all the pieces of one
tangram as the unit length. The square made from N complete
tangrams has a side of length sqrt(N). Such a side can be made up
from edges parallel to the edges of the unit square if N is a
perfect square, and sqrt(N) is an integer. It can be made up of
edges that run at angles of 45 degrees to the edges in a unit
square if N is twice a perfect square, and sqrt(2N) is an
integer. In other cases, the lengths of edges (all sums of
rational multiples of 1 and sqrt(2)) cannot possibly add to
sqrt(N).
From: Brasscat5@aol.com
Christine Francescani
Grade 10
Martin County High School
I started by making a reproduction of the tangram on my computer
and printing out a few. Once I had my very own tangrams I actually
started to think about which N's could be used to make a square. I
figured that if N is a perfect square, the tangrams can make a
square. Two tangrams can also make a square (see picture). I tried
3, but I don't think it would work. I found out that 8 works and
then figured out that if you can multiply N by 2 and it equals a
perfect square, then N can make a square (because a tangram can be
divided into 2 squares). So, any number of tangrams multiplied by
2 that equals a perfect square can make a square.
