INTRODUCTION
A. When a number is multiplied by itself, the resulting product is called a SQUARE NUMBER, or simply a SQUARE.
12 × 12 = 144  so 144 is a square number. 
35 × 35 = 1225  so 1225 is a square number. 
133 × 133 = 17,689  so 17689 is a square number. 
B. Sometimes a square is made up of digits that are all different, that is, it has “no repeats”. Such a square is called a distinctdigit square (DDS).
Example: 13 × 13 = 169; there are no repeated digits in 169,
so it is a distinctdigit square.But 21 × 21 = 441; since the 4 is repeated in 441, this is not
a distinctdigit square.
PROBLEM
You are to use your calculator to help you make a list of ten (10) distinctdigit squares. But–one more thing–they must all contain either 5 or 6 digits. That is, they should be “5place” or “6place” numbers.
Largest Number Squared
INTRODUCTION
If you multiply 142 by itself, what is the product? _________ If you multiply 781 by itself, what is the product? _________ Now look at your two answers. The first one was a 5place number, and the second one was a 6place number, right? (If not, you made a mistake somewhere. Do the wrong one(s) again.)
PROBLEM I
You now see than when you multiply a 3place number by itself, you might get a 5place or a 6place product.
Your problem is to use your calculator to find the largest 3place number that when multiplied by itself gives just a 5place product.
(Hint: The number is greater than 142.)
PROBLEM II
Compute these two products:
1022 × 1022 = ________ 
7803 × 7803 = ________ 
Do you see that the first product is a 7place number, and the second one is an 8place number? (If not, check your work as before.)
This time you are to find the largest 4place number which when multiplied by itself will still only make a 7place product.
(HINT: It is greater than 1022.)
PROBLEM III
Compute these two products:
17 × 17 = _______ 
83 × 83 = _______ 
Do you see that the first product is a 3place number, and the second one is a 4place number?
This time you are to find the largest 2place number which when multiplied by itself will still only make a 3place product.
(HINT: It is greater than 17.)
PROBLEM IV — The Brainbuster
You have done three problems with your calculator that were almost the same. Each time you had to find the largest number which
when multiplied by itself gave a product with an odd number of places,
right?
Now you will be asked to do the whole thing one more time–this is the BRAINBUSTER!
Find the largest fiveplace number which when multiplied by itself gives only a nineplace product.
But unfortunately, this time your calculator will not be able to help you; a 9place number is too big for the calculator’s display area.
However, things are not so bad if you will look at the answers you found for the first three problems. There is an important clue there that will tame this tough problem. Do you see it?
CLUE PATTERN:
The largest 3place product came from ______; 
The largest 5place product came from ______; 
The largest 7place product came from ______. 
SameDigit Pairs of DDSs
INTRODUCTION
In first section you found several squares that we called DDSs. (Remember: these are squares whose digits are “all different, no repeats”.)
In this section, we will explore something interesting about certain of those DDSs. Look at these squares:
37² = 1369 and 44² = 1936
Both 1369 and 1936 are DDSs, of course. BUT, there is one more thing that is strange: they both contain the same digits, just arranged in a different order.
There are many more cases like this. Before you start the exercise below, make sure you understand this idea by finding the squares for these two numbers: 32 and 49.
EXERCISES
In the groups of numbers below, two of them will give DDSs with the same digits, but arranged in a different order. The other numbers also produce DDSs, but do not have the same digits. Find the correct
pair in each group.

Below is given a large group of numbers that will give “samedigit pairs”, like you found above; some will not. Find the numbers that make this type of pair and put them together.
267  281  186  273  224 
213  282  286  226  214 
Once in a while we can find three or more DDSs that use the same digits. Look at this example:
36² = 1296 54² = 2916 96² = 9216
Do you see that all three squares contain the same digits, only in a different order. Now this is strange indeed! And it does not happen as oiften as was true for the samedigit pairs. But, as we will see, it can happen several times, if we are patient enough to look.
The following eleven numbers will produce DDSs that can be grouped into three samedigit families. Each family will have at least three members in it, maybe more. Can you separate all of them into their proper families?
181  148  154  128 
209  203  269  196 
191  302  178  . 
So far, all of our DDSs have been only 5place numbers. But the same thing can happen with 6place DDSs, too. And, would you believe it? There are even more pairs and familysized groups than you saw before.
Here are several numbers that will produce DDSs pairs or families. Can you separate them as you did before?
324  353  364  375 
403  405  445  463 
504  509  589  645 
661  708  843  905 
NOTE:
This piece was written by me and published in The Oregon Mathematics Teacher, Sept. 1978. At that time calculators with a 10digit display were not the common models available to students at the elementary or middle school levels. So the “Brainbuster” problem above needs to be adjusted to take that into account, or only permit the use of 8digit models while
doing this activity.