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.

I notice that the two solutions so far have used abc (base N) to denote P, and cycled alphabetically to get Q as bca and R as cab. If the cycling is done in the opposite sense, so that P is abc, Q is cab and R is bca (base N) then by using the same basic methods, further solutions can be found that are closely related to those already submitted.

Using Steve's notation, N = 4 + 3k (k = 1, 2, 3..) with a = 1,..k in each case, the other base N digits are now given by

b = a + (2N + 1)/3   and   c = a + (N + 2)/3

Using Charlie's method of listing, these further solutions look like this...

N          a  b  c           P      Q      R      <o:p></o:p>

<o:p> </o:p>

7          1  6  4          95     209    323<o:p></o:p>

  
10         1  8  5          185    518    851  
10         2  9  6          296    629    962<o:p></o:p>

  
13         1  10 6          305    1037   1769 
13         2  11 7          488    1220   1952 
13         3  12 8          671    1403   2135<o:p></o:p>

 
16         1  12 7          455    1820   3185 
16         2  13 8          728    2093   3458 
16         3  14 9          1001   2366   3731 
16         4  15 10         1274   2639   4004<o:p></o:p>

 
19         1  14 8          635    2921   5207 
19         2  15 9          1016   3302   5588 
19         3  16 10         1397   3683   5969 
19         4  17 11         1778   4064   6350 
19         5  18 12         2159   4445   6731<o:p></o:p>

 
22         1  16 9          845    4394   7943 
22         2  17 10         1352   4901   8450 
22         3  18 11         1859   5408   8957 
22         4  19 12         2366   5915   9464 
22         5  20 13         2873   6422   9971 
22         6  21 14         3380   6929   10478<o:p></o:p>

25         1  18 10         1085   6293   11501
25         2  19 11         1736   6944   12152
25         3  20 12         2387   7595   12803
25         4  21 13         3038   8246   13454
25         5  22 14         3689   8897   14105
25         6  23 15         4340   9548   14756
25         7  24 16         4991   10199  15407<o:p></o:p>

28         1  20 11         1355   8672   15989
28         2  21 12         2168   9485   16802
28         3  22 13         2981   10298  17615
28         4  23 14         3794   11111  18428
28         5  24 15         4607   11924  19241
28         6  25 16         5420   12737  20054
28         7  26 17         6233   13550  20867
28         8  27 18         7046   14363  21680

 

Posted by Harry on 2009-06-10 00:49:08