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Going Cyclic With Arithmetic (Posted on 2009-06-08) Difficulty: 2 of 5
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.

No Solution Yet Submitted by K Sengupta    
Rating: 3.0000 (1 votes)

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Solution computer solution | Comment 1 of 4

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

 


  Posted by Charlie on 2009-06-08 13:54:28
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