Three 3-digit positive base N integers
P,
Q and
R, each with no leading zeroes and having the restriction
P <
Q <
R, are such that:
- Q is the arithmetic mean of P and R, and:
- P, Q and R can be derived from one another by cyclic permutation of digits.
Determine all possible positive integer values of N < 30 for which this is possible.
The program found this works for N = 7, 10, 13, 16, 19, 22, 25 and 28.
Decimal
N Digits P Q R Difference
7 1 3 6 76 190 304 114 114
10 1 4 8 148 481 814 333 333
10 2 5 9 259 592 925 333 333
13 1 5 10 244 976 1708 732 732
13 2 6 11 427 1159 1891 732 732
13 3 7 12 610 1342 2074 732 732
16 1 6 12 364 1729 3094 1365 1365
16 2 7 13 637 2002 3367 1365 1365
16 3 8 14 910 2275 3640 1365 1365
16 4 9 15 1183 2548 3913 1365 1365
19 1 7 14 508 2794 5080 2286 2286
19 2 8 15 889 3175 5461 2286 2286
19 3 9 16 1270 3556 5842 2286 2286
19 4 10 17 1651 3937 6223 2286 2286
19 5 11 18 2032 4318 6604 2286 2286
22 1 8 16 676 4225 7774 3549 3549
22 2 9 17 1183 4732 8281 3549 3549
22 3 10 18 1690 5239 8788 3549 3549
22 4 11 19 2197 5746 9295 3549 3549
22 5 12 20 2704 6253 9802 3549 3549
22 6 13 21 3211 6760 10309 3549 3549
25 1 9 18 868 6076 11284 5208 5208
25 2 10 19 1519 6727 11935 5208 5208
25 3 11 20 2170 7378 12586 5208 5208
25 4 12 21 2821 8029 13237 5208 5208
25 5 13 22 3472 8680 13888 5208 5208
25 6 14 23 4123 9331 14539 5208 5208
25 7 15 24 4774 9982 15190 5208 5208
28 1 10 20 1084 8401 15718 7317 7317
28 2 11 21 1897 9214 16531 7317 7317
28 3 12 22 2710 10027 17344 7317 7317
28 4 13 23 3523 10840 18157 7317 7317
28 5 14 24 4336 11653 18970 7317 7317
28 6 15 25 5149 12466 19783 7317 7317
28 7 16 26 5962 13279 20596 7317 7317
28 8 17 27 6775 14092 21409 7317 7317
DEFDBL A-Z
FOR n = 2 TO 30
FOR d1 = 1 TO n - 1
FOR d2 = d1 TO n - 1
FOR d3 = d2 TO n - 1
n1 = d1 * n * n + d2 * n + d3
n2 = d2 * n * n + d3 * n + d1
n3 = d3 * n * n + d1 * n + d2
IF n3 < n2 THEN SWAP n2, n3
IF n3 - n2 = n2 - n1 AND n2 > n1 THEN
IF n > prevN THEN PRINT
PRINT n, d1; d2; d3, n1; n2; n3, n3 - n2; n2 - n1
prevN = n
END IF
NEXT
NEXT
NEXT
NEXT
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Posted by Charlie
on 2009-06-08 13:54:28 |
computer solution
