Pattern excerpts from *Mathematical Recreations* books

From *“Mathematical Recreations & Essays”*

by W.W.Rouse Ball & H.S.M. Coxeter, Twelfth Ed. p. 13,14.

Find all numbers which are integral multiples of their reversals.

Answer (Sloane's A031877) : for instance, among numbers of four digits,

8712 = 4 x 2178

and

9801 = 9 x 1089

Numbers that are integer multiples of their reversals are called **palintiples**.

Dan Hoey made a study and published his Solution to the /arithmetic/digits/palintiples problem.

From *“Mathematical Magic Show”* by Martin Gardner, page 211

Any number of **9**'s can be inserted in the middle of each number

to obtain larger (but dull) numbers with the same property;

for instance, 21999978 x 4 = 87999912.

Larger numbers can also be fabricated by repeating each fourdigit number:

thus, 2178 2178 2178 x 4 = 8712 8712 8712

and 1089 1089 1089 x 9 = 9801 9801 9801.

Of course numbers such as 21999978 may also be repeated to produce reversible numbers.

__Some considerations__

1089 is the square of a palindrome namely ( **33** )

9801 is the square of a palindrome namely ( **99** )

9801 – 1089 equals 8712 which is the first example !

8712 – 2178 equals 6534

6534 – 4356 equals 2178. The circle is closed !

From *“Figuring - The Joy of Numbers”* by Shakuntala Devi, page 70 and 122

"Numbers made up only of threes have a special pattern of squares"

33 ^{2} = 1089

333 ^{2} = 110889

3333 ^{2} = 11108889

33333 ^{2} = 1111088889

333333 ^{2} = 111110888889

Note that 33 equals 1! + 2! + 3! + 4!

and that 33 equals 1^{4} + 2^{5}

"The number 1089 has some peculiar traits.

For instance look at the pattern that is formed when it is multiplied by the numbers 1 to 9 :"

1089 x 1 = 1089 --- 9801 = 1089 x 9

1089 x 2 = 2178 --- 8712 = 1089 x 8

1089 x 3 = 3267 --- 7623 = 1089 x 7

1089 x 4 = 4356 --- 6534 = 1089 x 6

1089 x 5 = **5445**

The following comes from Shakuntala Devi's book

*“Figuring - The Joy of Numbers”* page 123 :

Multiplying the number 12345679 by the __multiples of 9__,

gives a curious set of palindromic patterns :

12345679 x 9 = **111111111**

12345679 x 18 = **222222222**

12345679 x 27 = **333333333**

and so on.

Multiplying by **999999999** itself gives a number which is a mirror image of itself

12345679 x **999999999** = **12345678987654321**

incidentally, this product is a perfect square. It's also Palindromic Square [**92**]

**111111111 **^{2} = **12345678987654321**

Many people find the above square quite useless but nevertheless fascinating !