**Palindromic Products of Non palindromic Integers and their Reversals** are defined and calculated by this extraordinary intricate

and excruciatingly complex formula. So, this line is for experts only _{}

base x esab

Here is a gem from Carlos Rivera's puzzle website ( Source Puzzle 52 ).

A 3-digit number trick.

Everybody must have heard of the following number trick that involves 'reverse & add'-ing

the digits of a random 3-digit number. If not here your chance to catch up.

I'm grateful to Mitch Beck for making me aware of
this property of 1089, a truly funny number.

Take any three integers from zero to nine, then subtract its reversal.

Then, if the difference is positive, add its reversal.

If the difference is negative, then subtract its reversal.

NO MATTER WHAT 3-DIGIT INTEGERS YOU BEGIN WITH, THE FINAL ANSWER IS ALWAYS 1089 !

856 159 872 - 658 - 951 - 278 ------ ------ ------ 198 - 792 594 + 891 - 297 + 495 ------ ------ ------ 1089 -1089 1089

On the following page 64 Beiler gives a __mathematical proof__ why this operation always works.

It might be worthwhile to order this book as it is almost a bible for number theory enthousiasts.

[ *September 10, 2002* ]

Sathiya Subramanian (email) from Redwood City, California

struggled with what he thought is a counterexample of the above 3-digit number trick.

He used integer 758 to start with ¬

The trick here is to keep using three digits even if758

– 857

--------

– 99

– 99

--------

– 189 ??

appears in the result ! Sathiya's example thus becomes ¬

A very sharp observation nevertheless. Thanks!758

– 857

--------

– 099

– 990

--------

– 1089 !!

When an integer and its reversal are unequal, their product is never a square except when both are squares.

Some observations about Palindromic Products of Integers & their Reversals.

[See table at the end of this page]

The basenumbers are made up of a combination of only three digits namely 0, 1 and 2 !

Although it is not the case, one could mistakenly believe that we are dealing with numbers in base 3 !!

Is someone prepared to find out why that is ? A mathematical proof will be much appreciated.

[ *August 9, 2000* ]

Henry Bottomley (email) cannot quite give a proof yet, but his explanation is no less interesting.

If the sum of the squares of the digits of n is less than 10 and n is not divisible by 10, then the product of n and the reversal of n is a palindrome. [Indeed extending this to any base k: if the sum of the squares of the base k digits of n is less than k and n is not divisible by k, then the product of n and the base k reversal of n is a base k palindrome.] The reason for this is: (i) any (perhaps partial) sum of pairwise products of digits of n is less than or equal to the sum of the square of the digits of n, which in turn is less than 10 [or k]; (ii) therefore there are no carries when calculating the digits of the product of n and its reversal; (iii) since multiplication and addition are commutative, this means that each digit of the product is equal to the digit an equal distance the other side of the middle of the product; (iv) and since n is not a multiple of 10 [or k] there are no difficulties with final zeros in the product. To take your example, the digits of base*esab are (before any carries): b*e; b*s+a*e; b*a+a*s+s*e; b*b+a*a+s*s+e*e; a*b+s*a+e*s; s*b+e*a; e*b All of these are less or equal to b*b+a*a+s*s+e*e which is less than 10 [or k], so there are no carries. So this appears to be a palindrome. Since e is not zero (and neither is b), e*b is not zero, and so indeed this is a palindrome. If the condition is not met, palindromes are (probably) impossible, because of the carries. So the possibilities for digits are:

- 3;
- two 2s, with one or zero 1s, and as many 0s as required (but not as a final digit);
- one 2, with from zero through to five 1s, and as many 0s as required (but not as a final digit);
- from two through to nine 1s, and as many 0s as required (but not as a final digit);
- 1;
- 0 (as a special case).

Explorations into the reversal world

The table lists all the palindromes only up to length 9. Larger ones are very easy to find.

Per given length their total number grows exponentially.

Length of Total genuine basenumber n x n_reversed 1 0 2 1 3 3 4 10 5 19 6 44 7 ?

On the contrary. What about repunital palindromes except for the middle digit :

Here is another one from More Palindromic Products of Integer Sequences :

It

Fascinating digit-swapping occurs many times with e.g. these two reversals :

Here is a finite pattern starting with the very first one of the table :1 0 2 1 x 1 2 0 1 = 1 2 2 6 2 2 1 X X X X 1 0 1 2 x 2 1 0 1 = 2 1 2 6 2 1 2

2 x 2 = 4Infinite patterns are abundant here. But this one has something special ! Read on.

12 x 21 = 252

112 x 211 = 23632

1112 x 2111 = 2347432

11112 x 21111 = 234585432

111112 x 211111 = 23456965432

All the first terms of the above pattern when added together with the square of their previous (less one) numbers

( see Quasi-Over-Squares ) also produce palindromes :

David Wilson found out an interesting palindrome-related fact

Article emailed at [ *Sun, 08 Mar 1998* ]

The number 1089 (the square of palindrome33) is interesting because it reverses when multiplied by 9: 9 * 1089 = 9801. Well, it turns out that multiplication by 9 reverses a positive integer if and only if that integer is of the form 99*p, wherepis a palindrome consisting of only the digits 0 and 1, in which every run of 0's or 1's is at least two digits long. For instance, the palindrome11000111100011fits the bill forp. Then 99 *p= 1089010998901089, and we see that 9 * 1089010998901089 = 9801098990109801. Similarly, the set of numbers which reverses when multiplied by 4 are those of the form 198 *p, wherepis of the same above-described palindromic form. Finally, it is obvious that the palindromes themselves are reversed when multiplied by 1. This exhausts the posibilities; inbase 10, the only multipliers that can reverse a positive integer are 1, 4, or 9. There is some evidence that similar types of things happen in other bases. For instance, inbase 5, a number of the form 13 *p, wherepis any palindrome consisting of 0's and 1's is reversed when multiplied by 2.

Eric Weisstein added an interesting article to his Math Encyclopedia about Reversals which are integral multiples of themselves.

I cannot withhold the following equation althought the numbers are not palindromic :

[ Source *“Curious and Interesting Numbers”* by David Wells, page 158 ]

Neil Sloane's “Integer Sequences” Encyclopedia can be consulted online :P. De Geest] entries to the table.Click here to view some entries to the table about palindromes. |

Index Nr | Info | Basenumber | Length |
---|---|---|---|

Palindromic Product of Integers & their Reversals | Length | ||

Integer_{[Normal]} x Integer_{[Reversed]} | |||

Halted with length 9 (See also A048343 and A048344) | |||

33 | 20.012 x 21.002 | 5 | |

420.292.024 | 9 | ||

32 | 12.002 x 20.021 | 5 | |

240.292.042 | 9 | ||

31 | 11.121 x 12.111 | 5 | |

134.686.431 | 9 | ||

30 | 11.112 x 21.111 | 5 | |

234.585.432 | 9 | ||

29 | 11.102 x 20.111 | 5 | |

223.272.322 | 9 | ||

28 | 11.021 x 12.011 | 5 | |

132.373.231 | 9 | ||

27 | 11.012 x 21.011 | 5 | |

231.373.132 | 9 | ||

26 | 11.002 x 20.011 | 5 | |

220.161.022 | 9 | ||

25 | 10.211 x 11.201 | 5 | |

114.373.411 | 9 | ||

24 | 10.202 x 20.201 | 5 | |

206.090.602 | 9 | ||

23 | 10.121 x 12.101 | 5 | |

122.474.221 | 9 | ||

22 | 10.112 x 21.101 | 5 | |

213.373.312 | 9 | ||

21 | 10.111 x 11.101 | 5 | |

112.242.211 | 9 | ||

20 | 10.102 x 20.101 | 5 | |

203.060.302 | 9 | ||

19 | 10.022 x 22.001 | 5 | |

220.494.022 | 9 | ||

18 | 10.021 x 12.001 | 5 | |

120.262.021 | 9 | ||

17 | 10.012 x 21.001 | 5 | |

210.262.012 | 9 | ||

16 | 10.011 x 11.001 | 5 | |

110.131.011 | 9 | ||

15 | 10.002 x 20.001 | 5 | |

200.050.002 | 9 | ||

14 | 2.012 x 2.102 | 4 | |

4.229.224 | 7 | ||

13 | 1.202 x 2.021 | 4 | |

2.429.242 | 7 | ||

12 | 1.121 x 1.211 | 4 | |

1.357.531 | 7 | ||

11 | 1.112 x 2.111 | 4 | |

2.347.432 | 7 | ||

10 | 1.102 x 2.011 | 4 | |

2.216.122 | 7 | ||

9 | 1.022 x 2.201 | 4 | |

2.249.422 | 7 | ||

8 | 1.021 x 1.201 | 4 | |

1.226.221 | 7 | ||

7 | 1.012 x 2.101 | 4 | |

2.126.212 | 7 | ||

6 | 1.011 x 1.101 | 4 | |

1.113.111 | 7 | ||

5 | 1.002 x 2.001 | 4 | |

2.005.002 | 7 | ||

4 | 122 x 221 | 3 | |

26.962 | 5 | ||

3 | 112 x 211 | 3 | |

23.632 | 5 | ||

2 | 102 x 201 | 3 | |

20.502 | 5 | ||

1 | 12 x 21 | 2 | |

252 | 3 |

David W. Wilson (email) found out an interesting palindrome-related fact - go to topic

Henry Bottomley (email) gives an explanation why the basenumbers are composed

only of digits 0, 1 and 2 - go to topic

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