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**