Welcome 1992!Author(s): Mary AndrewsSource: Mathematics in School, Vol. 21, No. 3 (May, 1992), p. 24Published by: The Mathematical AssociationStable URL: http://www.jstor.org/stable/30214880 .

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WELCOME

1

9

9

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* by Mary Andrews

Much has been made of 1991, its palindromic property having encouraged the study of these numbers even to the extent of Don Stewart's list of twenty-two investigations in September Mathematics in School.

So, will 1992 be a come-down? It is not palindromic ... or is it?

3 x 8 x 83 = 1992. We might call it "factor palindromic". This is not a

common occurrence: a factor palindromic year of this type has happened once before in my lifetime, but not in that of a school-age person. It will happen again soon, though, and some pupils MAY live to see another such year. I leave the reader to work out which years I have in mind, but look more carefully at these factor palindromes.

First the "3 family" of which 1992 is a member: 3 x 1 x 13 = 39 3 x 2 x 23 = 138 3 x 3 x 33 = 297 3 x 4 x 43 = 516 3x5 x53= 795 3 x 6 x 63 = 1134 3 x 7 x 73 = 1533 3 x 8 x 83 = 1992 3 x 9 x 93 = 2511

The sequence 39,138,297, ... 1992,2511 has first differences 99,159, ... and second differences 60,60. They are numbers of the form 3n(10n + 3) n = 1,2, ... 9. (Of course values of n > 9 do not give factor palindromes ... ever?)

Using 4's to generate the pattern starting 4 x 1 x 14 = 56 we obtain a sequence with second difference 80, the numbers being of the form 4n(10n + 4). Consideration of the general term mn(l0n + m) with m = 1,2 ... 9 and n = 1,2, ... 9 and simplifying the expression for the differences of three successive terms will explain this common differ- ence of 20m.

Now let us consider other factor palindromes using two digits. Those of type 38 x 8 x 3 give no new numbers, as multiplication is commutative, but could lead to different groupings and new formulae. 3 x 88 x 3 gets us back to multiples of 11, of which we have had enough in 1991! Then there is 388 x 3 and 3 x 883 ....

Here is the 3-group for each:

311 x3= 933 322 x 3 = 966 333 x 3 = 999 344 x 3 = 1032 355 x 3= 1065

and so on going up in 33's

3x 113= 339 3 x 223 = 669 3 x 333 = 999 3 x 443 = 1329 3 x 553 = 1659 and so on going up in 330's

"It's boring!" to use a popular pupil expression. (Though there is a recent year hidden somewhere there.)

That leaves writing the figures as two two-digit numbers:

e.g. 31 x 13 = 403 32 x 23 = 736 33 x 33 = 1089 ...

with general term (10m + n)(10n + m) and with second difference 20. (Prove it!)

Then you might try 5-figure factor palindromes or .... Finally all of this involves enough "number-crunching"

to suggest that a computer might help. This Basic program gives all the factor palindromes related to 1992:

10 FOR M=1 TO 9 20 FOR N=1 TO 9 30 A=

10,N+ M

40 Y = MNA 50 PRINT Y; 60 NEXT N 70 NEXT M 80 END

and changes to lines 30 and 40 give other groups. Alternatively, a spread-sheet can be used: a VIEW-

SHEET printout is shown. The spreadsheet version has the advantage of showing the factor digits. It can easily be adapted for other factor palindrome groups, providing a good exercise in formula construction and displaying instant numbers for scrutiny.

m/n 1 2 3 4 5 6 7 8 9 1 11 42 93 164 255 366 497 648 819 2 24 88 192 336 520 744 1008 1312 1656 3 39 138 297 516 795 1134 1533 1992 2511 4 56 192 408 704 1080 1536 2072 2688 3384 5 75 250 525 900 1375 1950 2625 3400 4275 6 96 312 648 1104 1680 2376 3192 4128 5184 7 119 378 777 1316 1995 2814 3773 4872 6111 8 144 448 912 1536 2320 3264 4368 5632 7056 9 171 522 1053 1764 2655 3726 4977 6408 8019

24 Mathematics in School, May 1992

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