Distinct Digit Squares

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 distinct-digit square (DDS).

Example: 13 × 13 = 169; there are no repeated digits in 169,
so it is a distinct-digit square.

But 21 × 21 = 441; since the 4 is repeated in 441, this is not
a distinct-digit square.

PROBLEM

You are to use your calculator to help you make a list of ten (10) distinct-digit squares. But–one more thing–they must all contain either 5 or 6 digits. That is, they should be “5-place” or “6-place” 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?  _________
The first one was a 5-place number, and the second one was a
6-place 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 3-place number by itself, you might get a 5-place or a 6-place product.

Your problem is to use your calculator to find the largest 3-place number that when multiplied by itself gives just a 5-place 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 7-place number, and the second one is an 8-place number? (If not, check your work as before.)

This time you are to find the largest 4-place number which when multiplied by itself will still only make a 7-place 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 3-place number, and the second one is a 4-place number?

This time you are to find the largest 2-place number which when multiplied by itself will still only make a 3-place product.

(HINT: It is greater than 17.)

PROBLEM IVThe 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 five-place number which when multiplied by itself gives only a nine-place product.

But unfortunately, this time your calculator will not be able to help you; a 9-place 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 3-place product came from ______; The largest 5-place product came from ______; The largest 7-place product came from ______.

Same-Digit 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.

 144, 175, 174 305, 153, 198 136, 228, 267, 309 233, 193, 305, 172 152, 142, 118, 179, 147

Below is given a large group of numbers that will give “same-digit 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 same-digit 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 same-digit 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 5-place numbers. But the same thing can happen with 6-place DDSs, too. And, would you believe it? There are even more pairs and family-sized 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 10-digit 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 8-digit models while
doing this activity.