Re: Enigma 1445 - Count in German

| Richard Heathfield wrote:
| Chappy said:
|> Enigma 1445 - Count in German
|> New Scientist magazine, 2 June 2007.
|> by Richard England.
|>
|> In the following statements digits have been
|> consistently replaced by capital letters,
|> different letters being used for different
|> digits.
|>
|> ZWEI is a prime number, DREI is a triangular
|> number, VIER is a perfect square.
|>
|> Triangular numbers are those that fit the
|> formula n(n+1)/2, like 1, 3, 6 and 10.
|>
|> What are the numbers represented by (a) DREI
|> and (b) VIER?

| There are 520 primes with exactly four discrete digits - too many to

I get 510 primes. _____________________________Gerard S.



| list here. There are 49 triangular numbers with exactly four discrete
| digits. They are:
|
| 1035 1275 1326 1378 1485 1540 1596 1653 1830 1953
| 2016 2145 2346 2415 2485 2701 2850 3081 3160 3240
| 3486 3570 3741 3916 4095 4186 4278 4371 4560 4753
| 4851 4950 5460 5671 6105 6328 6903 7021 7140 7260
| 7381 7503 8256 9045 9180 9316 9453 9730 9870
|
| There are just 36 perfect squares with exactly four discrete digits.
| They are:
| 1024 1089 1296 1369 1764 1849 1936 2304 2401 2601
| 2704 2809 2916 3025 3249 3481 3721 4096 4356 4761
| 5041 5184 5329 5476 6084 6241 6724 7056 7396 7569
| 7921 8649 9025 9216 9604 9801
|
| Here are those two lists again, shorn of the first digit, with the
| second list's digits reversed and each list re-sorted and uniquified
| (each list is a set of candidates for R E I in that order):
|
| 016 021 035 045 081 095 105 140 145 160
| 180 186 240 256 260 275 278 316 326 328
| 346 371 378 381 415 453 460 485 486 503
| 540 560 570 596 653 671 701 730 741 753
| 830 850 851 870 903 916 950 953
|
| 104 106 108 127 129 140 142 167 184 403
| 406 407 420 427 467 480 481 520 612 619
| 639 650 653 674 690 692 693 908 923 942
| 946 948 963 965 980
|
| We need the intersection of these two lists, which is:
|
| 140 653
|
| Not too many candidates after all, it seems.
|
| Okay, in our triangles list, these are 1653 and 7140, whereas in the
| squares list they are 4356 and 5041. We need a prime ending in 40 or
| 53, then. Clearly we won't find one ending in 40, so that eliminates
| the 7140/5041 pair, leaving 1653 and 4356 as the answers to (a) and (b)
| respectively. Incidentally, ZWEI is one of these: 2053, 2753, 2953,
| 6053, 7253, 7853, 8053, 8753
|
| --
| Richard Heathfield <http://www.cpax.org.uk>
| Email: -www. +rjh@
| Google users: <http://www.cpax.org.uk/prg/writings/googly.php>
| "Usenet is a strange place" - dmr 29 July 1999


.

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