It is a well known fact that if you permute the digits of a number the difference will be a multiple of 9.

Define the sequence D, where D(n) is the smallest positive value that can be increased by 9n through a permutation of its digits. No leading zeroes are allowed so the first term is D(1)=12 not 10

1) Find the next 14 terms of D.

2) Note D(8) is the greatest n with two digits. What is the greatest n with 3, 4, 5, ... digits?

3) There are some numbers a, b such that a≠b but D(a)=D(b). Prove there are infinitely many such pairs.

4) Sometimes D(n)>9n and sometimes D(n)<9n. Prove that both cases happen an infinity of times.
5) Are there any values of n such that D(n)=9n?

  n    9*n    D(n)
  2     18     13
  3     27     14
  4     36     15
  5     45     16
  6     54     17
  7     63     18
  8     72     19
  9     81    109
 10     90    120
 11     99    102
 12    108    102
 13    117    124
 14    126    125
 15    135    126
 16    144    127
 17    153    128
 18    162    129
 19    171    130
 20    180    130
 21    189    123
 22    198    103
 23    207    103
 24    216    135
 25    225    136
 26    234    137
 27    243    138
 28    252    139
 29    261    140
 30    270    140
 31    279    134
 32    288    124
 33    297    104
 34    306    104
 35    315    146
 36    324    147
 37    333    148
 38    342    149
 39    351    150
 40    360    150
 41    369    145
 42    378    135
 43    387    125
 44    396    105
 45    405    105
 46    414    157
 47    423    158
 48    432    159
 49    441    160
 50    450    160 
 51    459    156
 52    468    146
 53    477    136
 54    486    126
 55    495    106
 

Note that n in the program is not the n above, and the output was sorted for presentation above:
 
 DIM taken(200)
 FOR n = 11 TO 99999
   ns$ = LTRIM$(STR$(n))
   FOR add = 18 TO 55 * 9 STEP 9
     mult = add / 9
     IF taken(mult) = 0 THEN
       i = n + add
       ist$ = LTRIM$(STR$(i))
       IF LEN(ist$) = LEN(ns$) THEN
         good = 1
         FOR i = 1 TO LEN(ns$)
           ix = INSTR(ist$, MID$(ns$, i, 1))
           IF ix = 0 THEN good = 0: EXIT FOR
           ist$ = LEFT$(ist$, ix - 1) + MID$(ist$, ix + 1)
         NEXT
         IF good THEN
           PRINT USING "#######"; mult; add; n
           taken(mult) = 1
           ct = ct + 1
           IF ct MOD 40 = 0 THEN DO: LOOP UNTIL INKEY$ > "": PRINT
         END IF
       END IF
     END IF
   NEXT add
NEXT n

Edited on September 17, 2010, 5:29 pm

Posted by Charlie on 2010-09-17 17:28:18