perplexus.info :: Numbers : Vicious Circle

Vicious Circle (Posted on 2007-01-17)

Joe's latest puzzle for Penny consists of nine consecutively numbered pegs and a disc with 10 holes. The disc is free to rotate about its center.

The top position of the circle at any given time is marked by a fixed arrow pointer. A peg may be inserted in the disc in an empty hole adjacent to the arrow. If that hole isn't empty, she can't continue placing pegs.

The pegs are numbered, and after each one is placed, the disc is rotated that many positions clockwise. Penny has succeeded in placing all the pegs in the holes on the disc and has left an empty spot at the top adjacent to the arrow pointer, as shown in the following diagram. I've marked the empty spot that remains, with a hyphen:


                 |
                 V

                 -

       1                   9



 z                               x





 7                               2



       3                   8

                 y

But Penny doesn't remember in which of the positions marked x, y and z were the 4, 5 and 6 at the end of the game.

In what order did Penny place the pegs?

Submitted by Charlie

Rating: 5.0000 (1 votes)

Solution: (Hide)

We know that the sequence included 8, 3, 2, 9, and that x followed the 9. We also know that 7 was the last peg placed, as that left at the top the hole that's currently there. So in the following program, 8 stands for the whole sequence 8329:

DECLARE SUB place (p$) DECLARE SUB permute (a$) CLS DIM SHARED d$, flag a$ = "14568": h$ = a$ DO d$ = SPACE$(10): flag = 0 FOR i = 1 TO 5 IF MID$(a$, i, 1) = "8" THEN place "8" place "3" place "2" place "9" ELSE place MID$(a$, i, 1) END IF IF flag THEN EXIT FOR NEXT place "7" IF flag = 0 THEN PRINT a$; " "; d$ permute a$ LOOP UNTIL a$ = h$ END SUB place (p$) IF MID$(d$, 1, 1) > " " THEN flag = 1: EXIT SUB MID$(d$, 1, 1) = p$ rot = VAL(p$) psn = 11 - rot d$ = MID$(d$, psn) + LEFT$(d$, psn - 1) END SUB

The results show as:

placement final pos | V 56148 63 4159728 56841 694285371 61485 952863741 65841 942863715

where the left set of numbers is the sequence of placement and the right set shows the result, starting with the top and going clockwise.
The first two rows leave a 6 at the top, rather than a blank. Only the third line matches the numbered positions, and so the sequence of placed pegs was 6, 1, 4, 8, 3, 2, 9, 5, 7 (since 8 represents 8329, and 7 is understood at the end). By the way, x=5, y=6 and z=4, shown in positions on the right side of line 3.
From Enigma No. 1421 by Bob Walker, New Scientist, 9 December 2006.

Comments: ( You must be logged in to post comments.)

Subject	Author	Date
Puzzle Thoughts	K Sengupta	2023-07-16 02:52:13
Solution	Penny	2007-01-20 23:26:46
I thiiiink	TamTam	2007-01-17 15:01:23

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