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Building a ruler (Posted on 2008-11-14) Difficulty: 2 of 5
I have an unmarked ruler (AD) of length 6cm. Making two marks in it, one (B) at 1cm from the left end and other (C) at 2cm from the right end, Im able to measure any integer length from 1 to 6 cm:
       +----+-------------+--------+
       A    B             C        D
AB = 1cm / CD = 2cm / BC = 3cm / AC = 4cm / BD = 5cm / AD = 6cm.

If I have an unmarked ruler of length 14cm, what is the minimum number of marks, and where do I have to make them, in order to be able to measure any integer length from 1 to 14cm?

See The Solution Submitted by pcbouhid    
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Solution solution | Comment 2 of 10 |
(In reply to Catch? by ed bottemiller)

I don't think there was a catch.

I also origianally got 130 solutions but added a history internal array so I could weed out solutions that were the reverse of previous solutions.  It turns out (with hindsight) that I could have accomplished that by requiring a mark at 1; I didn't know ahead of time that no solution depends on having an internal distance of 1 cm.

The 65 solutions with 5 marks not counting their mirror images:

1   2   3   4   9
1   2   3   6  10
1   2   3   7  10
1   2   3   7  11
1   2   3   8  10
1   2   3   8  12
1   2   3   9  10
1   2   3   9  13
1   2   4   6  11
1   2   4   6  13
1   2   4   8  13
1   2   5   6  12
1   2   5   8  11
1   2   5   8  12
1   2   5   9  11
1   2   6   7  11
1   2   6   8  11
1   2   6   9  11
1   2   6   9  12
1   2   6  10  13
1   2   6  11  13
1   2   7   8  11
1   2   7   9  11
1   2   7  10  11
1   2   7  10  12
1   2   7  10  13
1   2   8   9  12
1   2   8  10  13
1   3   4   6  13
1   3   4   7  12
1   3   4   8  12
1   3   4   8  13
1   3   5   6  13
1   3   5   7  13
1   3   5   8  13
1   3   5  11  12
1   3   6   9  13
1   3   6  10  12
1   3   6  10  13
1   3   7   9  13
1   3   7  11  12
1   3   8   9  13
1   3   8  10  12
1   3   8  10  13
1   3   8  11  12
1   3   9  10  12
1   3   9  10  13
1   4   5   6  12
1   4   5   7  12
1   4   5   8  12
1   4   5  10  12
1   4   5  11  12
1   4   7   9  12
1   4   7  10  12
1   4   8   9  12
1   5   6  11  12
1   5   8  10  12
1   5   8  11  12
1   5   9  11  12
1   6   7  10  12
1   7   8  10  12
1   7   9  10  12
1   7   9  11  12
1   8   9  10  12
1   8   9  11  12

 

The program includes 7 "marks", counting two of which are the end points of the ruler.

DECLARE SUB place (which!)
DIM SHARED n, m(10), ct, h(200, 10)
m(1) = 0: m(2) = 14

CLS

FOR n = 4 TO 7
  REDIM SHARED used(14), lst(n)
  used(0) = 1: used(14) = 1
  place 3
NEXT

PRINT : PRINT ct

SUB place (which)
 IF which = 3 THEN st = 1:  ELSE st = m(which - 1) + 1
 FOR psn = st TO 14 - (n - which)
   IF used(psn) = 0 THEN
      used(psn) = 1
      m(which) = psn
      IF which = n THEN
        REDIM meas(14)
        FOR i = 1 TO n
         FOR j = 1 TO n
           dist = ABS(m(i) - m(j))
           meas(dist) = 1
         NEXT j
        NEXT i
        good = 1
        FOR i = 1 TO 14
         IF meas(i) = 0 THEN good = 0: EXIT FOR
        NEXT
        IF good THEN
          FOR i = 1 TO ct
            match = 1
            FOR j = 3 TO n
              IF m(j) <> 14 - h(i, n - j + 3) THEN match = 0: EXIT FOR
            NEXT
            IF match THEN good = 0: EXIT FOR
          NEXT i
          IF good THEN
            ct = ct + 1
            FOR i = 3 TO n
              PRINT USING "####"; m(i);
              h(ct, i) = m(i)
            NEXT
            PRINT
            IF ct MOD 40 = 0 THEN DO: LOOP UNTIL INKEY$ > "": PRINT
          END IF
        END IF
      ELSE
        place which + 1
      END IF
      used(psn) = 0
   END IF
 NEXT psn
END SUB

 


  Posted by Charlie on 2008-11-14 18:06:03
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