All about flooble | fun stuff | Get a free chatterbox | Free JavaScript | Avatars
 perplexus dot info

 Which multiple of 9? (Posted on 2010-09-17)
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?

 No Solution Yet Submitted by Jer Rating: 4.0000 (1 votes)

Comments: ( Back to comment list | You must be logged in to post comments.)
 A start: part 1 extended--D(2) thru D(55) | Comment 1 of 8
`  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
IF taken(mult) = 0 THEN
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 n

Edited on September 17, 2010, 5:29 pm
 Posted by Charlie on 2010-09-17 17:28:18

 Search: Search body:
Forums (0)