A multipan balance scale depicted below has one pan on the left and five on the right.

-+-       -+-  -+-  -+-  -+-  -+-
 |         |    |    |    |    |
 +----+----+----+----+----+----+
      |
      |

The pan on the left is a unit distance from the fulcrum. The pans on the right are 1, 2, 3, 4, and 5 units from the fulcrum. The balance scale is operated by placing an unknown weight on the left and placing some or all of a known set of weights on the pans on the right.

If the known weights are 1, 2, 3, 4 and 5, show that any integral weight 1 to 75 can be measured. What is the smallest measurement requiring 3 weights on the right? 4 weights? 5 weights?

If the restriction that only one weight can occupy a pan is added, show that any integral weight 1 to 55 can be measured. What is the smallest measurement requiring 3 weights on the right? 4 weights? 5 weights?

Assuming the materials of the balance scale itself are lighter on the right than on the left so that the scale balances when no weights are on either side:

In the table below, against each total weight to be weighed are shown the number of weights needed on the right, the number of weights needed if no more than one weight can be placed on a pan, and then the ways of accomplishing the weighing in a minimumn number of weights, with and without the 1-weight-per-pan restriction.  The latter descriptions list which pan each weight is to be placed in.

For example, the entry

      weights          
wgt  #  #   12345  12345 
------------------------
48   3  4     155   1254

indicates that the least number of weights for weighing 48 is 3 if allowed to place more than one weight on a pan, but 4 with the 1-weight-per-pan restriction. It's accomplished by placing the 3-weight on pan 1 and the 4- and 5-weights on pan 5. With the restriction, it's done by placing the 2 on pan 1, the 3 on pan 2, the 4 on pan 5 and the 5 on pan 4.

For the unrestricted case, the smallest weight requiring:

2 weights is 7
3 weights is 38
4 weights is 58
5 weights is 69

With the 1-weight-per-pan restriction, the smallest weight requiring:

2 weights is 7
3 weights is 36
4 weights is 48
5 weights is 55

      weights                           weights
wgt  #  # 12345  12345           wgt   #  #   12345  12345
----------------------           -------------------------
 1   1  1 1      1                39   3  3     144    352
 2   1  1  1      1               40   2  2      54     54
 3   1  1   1      1              41   2  2      45     45
 4   1  1    1      1             42   3  3     244    325
 5   1  1     1      1            43   3  3     154    154
 6   1  1   2      2              44   3  3     145    145
 7   2  2   11    21              45   2  3      55    425
 8   1  1    2      2             46   3  3     254    254
 9   1  1   3      3              47   3  3     245    245
10   1  1     2      2            48   3  4     155   1254
11   2  2   12     12             49   3  3     354    354
12   1  1    3      3             50   3  3     345    345
13   2  2    21     21            51   3  4     255   1354
14   2  2    12     12            52   3  4     454   1345
15   1  1     3      3            53   3  4     445   2354
16   1  1    4      4             54   3  4     355   2345
17   2  2    31     31            55   3  5     554  12345
18   2  2    22    1 3            56   3 9999   545
19   2  2    13     13            57   3 9999   455
20   1  1     4      4            58   4 9999  1545
21   2  2    41     41            59   4 9999  1455
22   2  2    32     32            60   3 9999   555
23   2  2    23     23            61   4 9999  2455
24   2  2    14     14            62   4 9999  1555
25   1  1     5      5            63   4 9999  3455
26   2  2    42     42            64   4 9999  2555
27   2  2    33    4 3            65   4 9999  4455
28   2  2    24     24            66   4 9999  3555
29   2  2    15     15            67   4 9999  5455
30   2  2    52     52            68   4 9999  4555
31   2  2    43     43            69   5 9999 14555
32   2  2    34     34            70   4 9999  5555
33   2  2    25     25            71   5 9999 15555
34   2  2   3 5    3 5            72   5 9999 25555
35   2  2    53     53            73   5 9999 35555
36   2  3    44    125            74   5 9999 45555
37   2  2    35     35            75   5 9999 55555
38   3  3   153    153

DIM minCt(75), minCt1(75), minarr(75, 5), minarr1(75, 5)
FOR i = 1 TO 75
  minCt(i) = 9999
  minCt1(i) = 9999
NEXT

OPEN "multipan.txt" FOR OUTPUT AS #2

FOR w1 = 0 TO 5
FOR w2 = 0 TO 5
FOR w3 = 0 TO 5
FOR w4 = 0 TO 5
FOR w5 = 0 TO 5

w = w1 + w2 * 2 + w3 * 3 + w4 * 4 + w5 * 5
PRINT #2, USING "### # # # # #"; w; w1; w2; w3; w4; w5;
REDIM pCt(5)
wCt = 0: ov1 = 0
FOR p = 1 TO 5
 IF w1 = p THEN pCt(p) = pCt(p) + 1: wCt = wCt + 1
 IF w2 = p THEN pCt(p) = pCt(p) + 1: wCt = wCt + 1
 IF w3 = p THEN pCt(p) = pCt(p) + 1: wCt = wCt + 1
 IF w4 = p THEN pCt(p) = pCt(p) + 1: wCt = wCt + 1
 IF w5 = p THEN pCt(p) = pCt(p) + 1: wCt = wCt + 1
 IF pCt(p) > 1 THEN ov1 = 1
NEXT
PRINT #2, wCt;
IF ov1 THEN
 PRINT #2, " >1 "
ELSE
 PRINT #2, "    "
END IF
IF wCt < minCt(w) THEN
 minCt(w) = wCt
 minarr(w, 1) = w1
 minarr(w, 2) = w2
 minarr(w, 3) = w3
 minarr(w, 4) = w4
 minarr(w, 5) = w5
END IF
IF wCt < minCt1(w) AND ov1 = 0 THEN

 minCt1(w) = wCt
 minarr1(w, 1) = w1
 minarr1(w, 2) = w2
 minarr1(w, 3) = w3
 minarr1(w, 4) = w4
 minarr1(w, 5) = w5

END IF
NEXT
NEXT
NEXT
NEXT
NEXT

CLOSE
CLS
col = 1
max1 = 1: max2 = 1
FOR i = 1 TO 75
  row = row + 1
  LOCATE row, col
  IF minCt(i) > max1 OR minCt1(i) > max2 THEN COLOR 14:  ELSE COLOR 7
  PRINT USING "### ###"; i; minCt(i); minCt1(i);
  COLOR 7
  FOR j = 1 TO 5
   IF minarr(i, j) = 0 THEN PRINT " "; :  ELSE PRINT LTRIM$(STR$(minarr(i, j)));
  NEXT
  PRINT "  ";
  FOR j = 1 TO 5
   IF minarr1(i, j) = 0 THEN PRINT " "; :  ELSE PRINT LTRIM$(STR$(minarr1(i, j)));
  NEXT

  IF minCt(i) > max1 THEN max1 = minCt(i)
  IF minCt1(i) > max2 THEN max2 = minCt1(i)
  IF row > 75 / 2 THEN col = 35: row = 0
NEXT
LOCATE 40, 1

 

Posted by Charlie on 2009-01-18 16:54:22