Re: Enigma 1432 - Roman and Arabic

Somebody claiming to be Chappy <petergregorychapman@xxxxxxxxxxx> wrote at
Thu, 26 Apr 2007 04:02:30 GMT:

[original problem preserved as spoiler space]

Enigma 1432 - Roman and Arabic
New Scientist magazine, 3 March 2007.
by Richard England.

XVI and LXIV are both Roman numeral
perfect squares. If you replace the
capital letters consistently with digits,
using a different digit for each letter,
you can translate XVI and LXIV into Arabic
numeral perfect squares, neither of which
starts with a zero.

Using the digital values of I, V, X and L
that you have discovered; (1) find another
Roman numeral perfect square that uses
only these letters (more than one but not
necessarily all of them) and translates
into an Arabic numeral perfect square;
(2) find a valid Roman numeral that uses
only these letters (more than one but not
necessarily all of them) and is not itself
a perfect square but translates into an
Arabic perfect square.

Send in the Roman numerals you found in
answer to (1) and (2) and the Arabic
numerals that each of them translates to.

Ciao,
Chappy.

Perfect squares have to end in 0, 1, 4, 5, 6, or 9. We note that XVI and
LXIV have the ones and tens places transposed. Possible final two digits
for squares:

00
01 21 41 61 81
04 24 44 64 84
25
16 36 56 76 96
09 29 49 69 89

It's possible for XVI and LXIV to end in 69 and 96, or 16 and 61. Looking
through three- and four-digit squares, we get possibilities of

XVI: 169 196 961
LXIV: 1296 1369 2916 4761 7396 7569 9216

The only ones that have a common digit for X are 2916 and 961. Therefore,
I = 1, V = 6, X = 9, and L = 2

For 1) obivously IV is Roman numerals for 4, and translates to 16. We see
above that 9216 is a perfect square, and translates to XLIV, which is a
valid roman numeral. So, the answers are:

1) IV, 16
2) XLIV, 9216

--
Ted <fedya at bestweb dot net>
Hmmm.... Eternal happiness for one dollar? [Pauses] On second thought,
I'd be happier *with* the dollar. --Montgomery Burns
<http://www.snpp.com/episodes/4F01.html>
.

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