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.

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.

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 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.)

     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:

     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:

     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 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!

     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?

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:

     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 "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:

     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: