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 Multipan Balance (Posted on 2009-01-18)
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?

 No Solution Yet Submitted by Brian Smith Rating: 3.0000 (1 votes)

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 computer solution | Comment 1 of 3

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                           weightswgt  #  # 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    24510   1  1     2      2            48   3  4     155   125411   2  2   12     12             49   3  3     354    35412   1  1    3      3             50   3  3     345    34513   2  2    21     21            51   3  4     255   135414   2  2    12     12            52   3  4     454   134515   1  1     3      3            53   3  4     445   235416   1  1    4      4             54   3  4     355   234517   2  2    31     31            55   3  5     554  1234518   2  2    22    1 3            56   3 9999   54519   2  2    13     13            57   3 9999   45520   1  1     4      4            58   4 9999  154521   2  2    41     41            59   4 9999  145522   2  2    32     32            60   3 9999   55523   2  2    23     23            61   4 9999  245524   2  2    14     14            62   4 9999  155525   1  1     5      5            63   4 9999  345526   2  2    42     42            64   4 9999  255527   2  2    33    4 3            65   4 9999  445528   2  2    24     24            66   4 9999  355529   2  2    15     15            67   4 9999  545530   2  2    52     52            68   4 9999  455531   2  2    43     43            69   5 9999 1455532   2  2    34     34            70   4 9999  555533   2  2    25     25            71   5 9999 1555534   2  2   3 5    3 5            72   5 9999 2555535   2  2    53     53            73   5 9999 3555536   2  3    44    125            74   5 9999 4555537   2  2    35     35            75   5 9999 5555538   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

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