Those Amazing Palindromes

by
Shareef Bacchus

It is not totally clear when palindromic numbers really came into being nor is it clear who deserves the credit for creating the numbers. Suffice it to say that the numbers have some fascinating properties that bear investigating.

Palindromes were first used in language to define words or lines that read the same backwards or forward. The word is of Greek origin coming from "palin dromo" which translates approximately to "to read back again." Examples of palindromes include:

and

"Roma tibi subito motibus ibit amor."

They have also been called Sotadics after their reputed inventor Sotades, a Greek poet who lived around 300BC. Palindromic words were also used in other languages including English. The longest palindrome in English probably is:

"Dog as a devil deified

Deified lived as a god"

Two other well known palindromes are

"Lewd did I live, evil I did dwel"

and Napoleon's famous reputed quotation:

"Able was I ere I saw Elba."

One of the most celebrated palindrome is the Greek word

NIYONANOMHMATAHMOMANDYIN.

The word means "wash my transgressions not my face".

Palindromic numbers like palindromic words or lines obey the same property of being the same whether they are read from left to right or vice-versa. Examples include:

484

123321

Palindromes could be formed from a number that is not a palindrome by adding the original number to the number formed by reversing the digits. For example:

38 + 83 = 121

which is a palindrome. Sometimes, however, more than one reversal is necessary. Thus,

49 + 94 = 143

which is not a palindrome. But

143 + 484

is a palindrome.

Writing in the Arithmetic Teacher of January 1985, Clarence J. Dockweiler in an article entitled "Palindromes and the laws of 11" made the following observation:

"If in the process of obtaining a palindrome, a sum with an even number of digits is obtained, the palindrome will be a multiple of 11."

He warned, however, that he was not sure that every number could be turned into a palindrome.

This researcher independently observed that palindromes with an even number of digits were divisible by 11 and set out to prove that all such palindromes, whether they arose from the reversal-sum process or not, were divisible by 11.

A general six-digit palindrome will be chosen but the results will be generalized to include all even numbered palindromes. The method of Induction will be used to verify the hypothesis.

HYPOTHESIS:

All palindromes with an even number of digits are divisible by 11.

PROOF:

Consider the palindrome

baccab

This could be written as

The following spreadsheet shows what happens when we divide palindromes with an even number of digits by 11. The palindromes were created by using the relationship established earlier. Six-digit palindromes can be created by the relationship:

where a,b,c all lie between 0 and 9.

In the first, second, and third columns we have values of a,b,and c. In the fourth column we have the palindromes formed by using the relationship above and in the fifth column we have the results obtained when we divide the palindromes by 11. Note that we have an integral answer every time.

It is also interesting to note that those palindromes with digits that increase from the left and then decrease appropriately as we get to the middle digit(s) have palindromic quotients when they are divided by 11!

```
1             1  1             111111        10101
1             1  2             112211        10201
1             1  3             113311        10301
1             1  4             114411        10401
1             1  5             115511        10501
1             1  6             116611        10601
1             1  7             117711        10701
1             1  8             118811        10801
1             1  9             119911        10901
1             2  1             211112        19192
1             2  2             212212        19292
1             2  3             213312        19392
1             2  4             214412        19492
1             2  5             215512        19592
1             2  6             216612        19692
1             2  7             217712        19792
1             2  8             218812        19892
1             2  9             219912        19992
1             3  1             311113        28283
1             3  2             312213        28383
1             3  3             313313        28483
1             3  4             314413        28583
1             3  5             315513        28683
1             3  6             316613        28783
1             3  7             317713        28883
1             3  8             318813        28983
1             3  9             319913        29083
1             3  1             311113        28283
1             3  2             312213        28383
1             3  3             313313        28483
1             3  4             314413        28583
1             3  5             315513        28683
1             3  6             316613        28783
1             3  7             317713        28883
1             3  8             318813        28983
1             3  9             319913        29083
1             4  1             411114        37374
2             4  1             421124        38284
3             4  1             431134        39194
4             4  1             441144        40104
5             4  1             451154        41014
6             4  1             461164        41924
7             4  1             471174        42834
8             4  1             481184        43744
9             4  1             491194        44654
5             1  1             151151        13741
5             1  2             152251        13841
5             1  3             153351        13941
5             1  4             154451        14041
5             1  5             155551        14141
5             1  6             156651        14241
5             1  7             157751        14341
5             1  8             158851        14441
5             1  9             159951        14541
6             1  1             161161        14651
6             1  2             162261        14751
6             1  3             163361        14851
6             1  4             164461        14951
6             1  5             165561        15051
6             1  6             166661        15151
6             1  7             167761        15251
6             1  8             168861        15351
6             1  9             169961        15451
7             1  1             171171        15561
7             1  2             172271        15661
7             1  3             173371        15761
7             1  4             174471        15861
7             1  5             175571        15961
7             1  6             176671        16061
7             1  7             177771        16161
7             1  8             178871        16261
7             1  9             179971        16361
8             1  1             181181        16471
8             1  2             182281        16571
8             1  3             183381        16671
8             1  4             184481        16771
8             1  5             185581        16871
8             1  6             186681        16971
8             1  7             187781        17071
8             1  8             188881        17171
8             1  9             189981        17271
9             1  1             191191        17381
9             1  2             192291        17481
9             1  3             193391        17581
9             1  4             194491        17681
9             1  5             195591        17781
9             1  6             196691        17881
9             1  7             197791        17981
9             1  8             198891        18081
9             1  9             199991        18181
1             5  1             511115        46465
1             5  2             512215        46565
1             5  3             513315        46665
1             5  4             514415        46765
1             5  5             515515        46865
1             5  6             516615        46965
1             5  7             517715        47065
1             5  8             518815        47165
1             5  9             519915        47265
1             6  1             611116        55556
1             6  2             612216        55656
1             6  3             613316        55756
1             6  4             614416        55856
1             6  5             615516        55956
1             6  6             616616        56056
1             6  7             617716        56156
1             6  8             618816        56256
1             6  9             619916        56356
1             7  1             711117        64647
1             7  2             712217        64747
1             7  3             713317        64847
1             7  4             714417        64947
1             7  5             715517        65047
1             7  6             716617        65147
1             7  7             717717        65247
1             7  8             718817        65347
1             7  9             719917        65447
```

REFERENCES

Benet's Readers Enclycopaedia, (1987). Harper and Row

Dockweiler, C.J. (1985). Palindromes and the "Law of 11". Arithmetic Teacher, 32(5)