You have to sort five weights weighing 51g, 52g, 53g, 54g, and 55g. You have a balance scale with which you can compare the weights. But after solving so many sorting puzzles, it is starting to break down.
If the difference between two weights is greater than 1.5g, the scale will correctly determine which side is heavier. If the difference between the weights is less than 1.5g or equal, the scale will indicate the weights are equal.
Sort the weights in the smallest number of weighings.
First, per information theory, there are 120 possible orders for the five weights, from weight a being the lightest and e the heaviest to the other way around and all the permutations in between. The broken scale has three possible readings: left pan heave, right pan heavy, pans level. Thus the number of possible sequences of n readings is 3^n. The smallest power of 3 that's at least 120 is 3^5, so no solution could have fewer than 5 weighings. As a sidelight, it's probably true that this imperfect scale will require fewer weighings than a perfect scale. In the table of possible weighings below, the imperfect scale shows 1080 instances of the pans balancing, 960 of one side being heavier and 960 of the other side being heavier. A perfect scale would have 360 cases of balanced pans, 1320 unbalanced one way and 1320 the other way--less well distributed.
It remains to be seen if weighings can be arranged such that 5 weighings suffice with the imperfect scale. As a start, consider that there are only 25 weighings that make sense: ten of one weight against another and fifteen of two weights against two weights. Having different numbers of weights on the two pans will result only in the side with the larger number being heavier and doesn't tell you anything. With that in mind as a limitation, we can then just as well count the weights as 1 - 5, rather than 51-55, so long as we limit ourselves to equal numbers of weights on each side. A table of all possible weighings for each of the possible orders follows. The order of weights is shown at the left (51 g to 55 g), and the headings show the weighings, the content of one pan above the content of the other. The entries are either ^ or v and represent whether the total weight in the pan represented on top is lighter or heavier, respectively than the pan represented on bottom, with = shown if they differ by less than 1.5 g.
grams
55555
12345
a a a a b b b c c d ab ab ab ac ac ac ad ad ad ae ae ae bc bd be
b c d e c d e d e e cd ce de bd be de bc be ce bc bd cd de ce cd
abcde = ^ ^ ^ = ^ ^ = ^ = ^ ^ ^ ^ ^ ^ = ^ ^ = = = ^ ^ =
abced = ^ ^ ^ = ^ ^ ^ = = ^ ^ ^ ^ ^ ^ = = = = ^ ^ ^ = ^
abdce = ^ ^ ^ ^ = ^ = = ^ ^ ^ ^ = ^ ^ ^ ^ ^ = = = ^ ^ =
abdec = ^ ^ ^ ^ = ^ v = = ^ ^ ^ = = = ^ ^ ^ ^ = ^ = ^ ^
abecd = ^ ^ ^ ^ ^ = = = v ^ ^ ^ ^ = ^ = = = ^ ^ ^ ^ = ^
abedc = ^ ^ ^ ^ ^ = = v = ^ ^ ^ = = = ^ = ^ ^ ^ ^ = ^ ^
acbde ^ = ^ ^ = = ^ ^ ^ = ^ ^ ^ ^ ^ ^ = ^ ^ = = = ^ = v
acbed ^ = ^ ^ = ^ = ^ ^ = ^ ^ ^ ^ ^ ^ = = = = ^ ^ ^ v =
acdbe ^ = ^ ^ v = = = ^ ^ = ^ ^ ^ ^ ^ ^ ^ ^ = = = ^ = v
acdeb ^ = ^ ^ v v = = ^ = = = = ^ ^ ^ ^ ^ ^ ^ ^ = = v v
acebd ^ = ^ ^ v = = ^ = v ^ = ^ ^ ^ ^ = = = ^ ^ ^ ^ v =
acedb ^ = ^ ^ v = v ^ = = = = = ^ ^ ^ ^ ^ = ^ ^ ^ = v v
adbce ^ ^ = ^ = = ^ v = ^ ^ ^ ^ = ^ ^ ^ ^ ^ = = = = ^ v
adbec ^ ^ = ^ ^ = = v = ^ ^ ^ ^ = = = ^ ^ ^ ^ = ^ v ^ =
adcbe ^ ^ = ^ = v = = ^ ^ = ^ ^ ^ ^ ^ ^ ^ ^ = = = = ^ v
adceb ^ ^ = ^ v v = = = ^ = = = ^ ^ ^ ^ ^ ^ ^ ^ = v = v
adebc ^ ^ = ^ = v = v v = ^ ^ = = = = ^ ^ ^ ^ ^ ^ v ^ =
adecb ^ ^ = ^ = v v v = = = = = ^ ^ = ^ ^ ^ ^ ^ ^ v = v
aebcd ^ ^ ^ = = ^ = = v v ^ ^ ^ ^ = ^ = = = ^ ^ ^ = v ^
aebdc ^ ^ ^ = ^ = = = v v ^ ^ ^ = = = ^ = ^ ^ ^ ^ v = ^
aecbd ^ ^ ^ = = = v ^ = v ^ = ^ ^ ^ ^ = = = ^ ^ ^ = v ^
aecdb ^ ^ ^ = v = v = = v = = = ^ ^ ^ ^ ^ = ^ ^ ^ v v =
aedbc ^ ^ ^ = = = v v v = ^ ^ = = = = ^ ^ ^ ^ ^ ^ v = ^
aedcb ^ ^ ^ = = v v = v = = = = ^ ^ = ^ ^ ^ ^ ^ ^ v v =
bacde = = ^ ^ ^ ^ ^ = ^ = ^ ^ ^ = = ^ v = ^ v v = ^ ^ =
baced = = ^ ^ ^ ^ ^ ^ = = ^ ^ ^ = = ^ v v = v = ^ ^ = ^
badce = ^ = ^ ^ ^ ^ = = ^ ^ ^ ^ v = ^ = = ^ v v = ^ ^ =
badec = ^ = ^ ^ ^ ^ v = = ^ ^ ^ v v = = = ^ = v ^ = ^ ^
baecd = ^ ^ = ^ ^ ^ = = v ^ ^ ^ = v ^ v v = = = ^ ^ = ^
baedc = ^ ^ = ^ ^ ^ = v = ^ ^ ^ v v = = v ^ = = ^ = ^ ^
bcade v = = ^ = ^ ^ ^ ^ = ^ ^ ^ = = ^ v = = v v v ^ ^ =
bcaed v = ^ = = ^ ^ ^ ^ = ^ ^ ^ = = ^ v v v v = = ^ = ^
bcdae v v = = = ^ ^ = ^ ^ = ^ ^ v = ^ v = = v v v ^ ^ =
bcdea v v v = = ^ ^ = ^ = = = = v v = v v v v v v ^ ^ =
bcead v v = = = ^ ^ ^ = v ^ = ^ = v ^ v v v v = = ^ = ^
bceda v v = v = ^ ^ ^ = = = = = v v = v v v v v v ^ = ^
bdace v = = ^ ^ = ^ v = ^ ^ ^ ^ v = = = = ^ v v v ^ ^ =
bdaec v ^ = = ^ = ^ v = ^ ^ ^ ^ v v v = = ^ = v = = ^ ^
bdcae v = v = ^ = ^ = ^ ^ = ^ ^ v = = v = ^ v v v ^ ^ =
bdcea v v v = ^ = ^ = = ^ = = = v v v v v = v v v ^ ^ =
bdeac v = v = ^ = ^ v v = ^ ^ = v v v = v ^ = v = = ^ ^
bdeca v = v v ^ = ^ v = = = = = v v v v v = v v v = ^ ^
beacd v = ^ = ^ ^ = = v v ^ ^ ^ = v = v v v = = ^ ^ = ^
beadc v ^ = = ^ ^ = = v v ^ ^ ^ v v v = v = = = ^ = ^ ^
becad v = = v ^ ^ = ^ = v ^ = ^ = v = v v v v = ^ ^ = ^
becda v v = v ^ ^ = = = v = = = v v v v v v v v = ^ = ^
bedac v = = v ^ ^ = v v = ^ ^ = v v v = v = = v ^ = ^ ^
bedca v = v v ^ ^ = = v = = = = v v v v v v v v = = ^ ^
cabde = = ^ ^ v = ^ ^ ^ = = = ^ ^ ^ ^ v ^ = v = v ^ = v
cabed = = ^ ^ v ^ = ^ ^ = = = ^ ^ ^ ^ v = v v ^ = ^ v =
cadbe ^ = = ^ v = = ^ ^ ^ v = ^ ^ ^ ^ = ^ = v = v ^ = v
cadeb ^ = = ^ v v = ^ ^ = v v = ^ ^ ^ = ^ = = ^ v = v v
caebd ^ = ^ = v = = ^ ^ v = v ^ ^ ^ ^ v = v = ^ = ^ v =
caedb ^ = ^ = v = v ^ ^ = v v = ^ ^ ^ = ^ v = ^ = = v v
cbade = v = ^ = ^ ^ ^ ^ = = = ^ ^ ^ ^ v = = v v v ^ = v
cbaed = v ^ = = ^ ^ ^ ^ = = = ^ ^ ^ ^ v v v v = = ^ v =
cbdae v v = = = = ^ ^ ^ ^ v = ^ = ^ ^ v = = v v v ^ = v
cbdea v v v = = = ^ ^ ^ = v v = = = = v v v v v v ^ = v
cbead v v = = = ^ = ^ ^ v = v ^ ^ = ^ v v v v = = ^ v =
cbeda v v = v = ^ = ^ ^ = v v = = = = v v v v v v ^ v =
cdabe = v = ^ v v = = ^ ^ v = = ^ ^ ^ = ^ = v v v ^ = v
cdaeb ^ v = = v v = = ^ ^ v v v ^ ^ ^ = ^ = = = v = v v
cdbae = v v = v = ^ = ^ ^ v = = = ^ ^ v ^ = v v v ^ = v
cdbea v v v = v = = = ^ ^ v v v = = = v = v v v v ^ = v
cdeab = v v = v v v = ^ = v v v ^ ^ = = ^ v = = v = v v
cdeba = v v v v v = = ^ = v v v = = = v = v v v v = v v
ceabd = v ^ = v = v ^ = v = v = ^ ^ ^ v v v = ^ = ^ v =
ceadb ^ v = = v = v ^ = v v v v ^ ^ ^ = = v = ^ = = v v
cebad = v = v v ^ = ^ = v = v = ^ = ^ v v v v ^ = ^ v =
cebda v v = v v = = ^ = v v v v = = = v v v v = v ^ v =
cedab = v = v v v v ^ = = v v v ^ ^ = = = v = ^ v = v v
cedba = v v v v = v ^ = = v v v = = = v v v v = v = v v
dabce = ^ = ^ = v ^ v = ^ = ^ = v ^ = ^ ^ ^ = v v = ^ v
dabec = ^ = ^ ^ v = v = ^ = ^ = v = v ^ ^ ^ ^ v = v ^ =
dacbe ^ = = ^ = v = v ^ ^ v ^ = = ^ = ^ ^ ^ = v v = ^ v
daceb ^ = = ^ v v = v = ^ v = v = ^ = ^ ^ ^ ^ = v v = v
daebc ^ ^ = = = v = v v ^ = ^ v v = v ^ ^ ^ ^ = = v ^ =
daecb ^ ^ = = = v v v = ^ v = v = ^ v ^ ^ ^ ^ = = v = v
dbace = = v ^ ^ = ^ v = ^ = ^ = v = = ^ ^ ^ v v v = ^ v
dbaec = ^ v = ^ = ^ v = ^ = ^ = v v v ^ ^ ^ = v = v ^ =
dbcae v = v = = = ^ v ^ ^ v ^ = v = = = ^ ^ v v v = ^ v
dbcea v v v = = = ^ v = ^ v = v v v v = = = v v v = ^ v
dbeac v = v = ^ = = v v ^ = ^ v v v v ^ = ^ = v = v ^ =
dbeca v = v v ^ = = v = ^ v = v v v v = = = v v v v ^ =
dcabe = = v ^ v v = = ^ ^ v = = = ^ = ^ ^ ^ v v v = ^ v
dcaeb ^ = v = v v = = ^ ^ v v v = ^ = ^ ^ ^ = = v v = v
dcbae = v v = = v ^ = ^ ^ v = = v ^ = = ^ ^ v v v = ^ v
dcbea v v v = = v = = ^ ^ v v v v = v = = = v v v = ^ v
dceab = v v = v v v = = ^ v v v = ^ v ^ ^ = = = v v = v
dceba = v v v v v = = = ^ v v v v = v = = = v v v v = v
deabc = ^ v = = v v v v = = = v v v v ^ ^ ^ ^ = = v ^ =
deacb ^ = v = = v v v v = v v v = = v ^ ^ ^ ^ = = v = v
debac = = v v ^ v = v v = = = v v v v ^ = ^ ^ v = v ^ =
debca v = v v = v = v v = v v v v v v = = = = v v v ^ =
decab = = v v v v v v = = v v v = = v ^ ^ = ^ = v v = v
decba = v v v = v v v = = v v v v v v = = = = v v v = v
eabcd = ^ ^ = = ^ v = v v ^ = = ^ v = = v v ^ ^ ^ = v ^
eabdc = ^ ^ = ^ = v = v v ^ = = = v v ^ v = ^ ^ ^ v = ^
eacbd ^ = ^ = = = v ^ v v ^ v = ^ = = = v v ^ ^ ^ = v ^
eacdb ^ = ^ = v = v = v v = v v ^ = = ^ = v ^ ^ ^ v v =
eadbc ^ ^ = = = = v v v v ^ = v = v v ^ = = ^ ^ ^ v = ^
eadcb ^ ^ = = = v v = v v = v v ^ = v ^ = = ^ ^ ^ v v =
ebacd = = ^ v ^ ^ = = v v ^ = = = v = v v v ^ ^ ^ = v ^
ebadc = ^ = v ^ ^ = = v v ^ = = v v v = v = ^ ^ ^ v = ^
ebcad v = = v = ^ = ^ v v ^ v = = v = v v v = ^ ^ = v ^
ebcda v v = v = ^ = = v v = v v v v v v v v = = = = v ^
ebdac v = = v ^ = = v v v ^ = v v v v = v = ^ = ^ v = ^
ebdca v = v v ^ = = = v v = v v v v v v v v = = = v = ^
ecabd = = ^ v v = v ^ = v = v = ^ = = v v v ^ ^ ^ = v ^
ecadb ^ = = v v = v ^ = v v v v ^ = = = = v ^ ^ ^ v v =
ecbad = v = v = ^ v ^ = v = v = ^ v = v v v = ^ ^ = v ^
ecbda v v = v = = v ^ = v v v v = v v v v v = = = = v ^
ecdab = v = v v v v = = v v v v ^ = v = = v ^ ^ = v v =
ecdba = v v v v = v = = v v v v = v v v v v = = = v v =
edabc = ^ = v = v v v v = = = v v v v ^ = = ^ ^ ^ v = ^
edacb ^ = = v = v v v v = v v v = = v ^ = = ^ ^ ^ v v =
edbac = = v v ^ = v v v = = = v v v v ^ v = ^ = ^ v = ^
edbca v = v v = = v v v = v v v v v v = v v = = = v = ^
edcab = = v v v v v = v = v v v = = v ^ = v ^ ^ = v v =
edcba = v v v = v v = v = v v v v v v = v v = = = v v =
This was produced by
DECLARE SUB permute (a$)
DEF fnwt (x) = INSTR(a$, CHR$(96 + x))
'CLS
OPEN "brokscal.txt" FOR OUTPUT AS #2
PRINT #2, TAB(7);
FOR upper = 1 TO 4
FOR lower = upper + 1 TO 5
PRINT #2, CHR$(96 + upper); " ";
NEXT
NEXT
FOR upper1 = 1 TO 4
FOR upper2 = upper1 + 1 TO 5
FOR lower1 = upper1 + 1 TO 4
IF lower1 <> upper2 THEN
FOR lower2 = lower1 + 1 TO 5
IF lower2 <> upper2 THEN
PRINT #2, CHR$(96 + upper1); CHR$(96 + upper2); " ";
END IF
NEXT
END IF
NEXT
NEXT
NEXT
PRINT #2,
PRINT #2, TAB(7);
FOR upper = 1 TO 4
FOR lower = upper + 1 TO 5
PRINT #2, CHR$(96 + lower); " ";
NEXT
NEXT
FOR upper1 = 1 TO 4
FOR upper2 = upper1 + 1 TO 5
FOR lower1 = upper1 + 1 TO 4
IF lower1 <> upper2 THEN
FOR lower2 = lower1 + 1 TO 5
IF lower2 <> upper2 THEN
PRINT #2, CHR$(96 + lower1); CHR$(96 + lower2); " ";
END IF
NEXT
END IF
NEXT
NEXT
NEXT
PRINT #2,
a$ = "abcde": h$ = a$
DO
PRINT #2, a$;
FOR upper = 1 TO 4
FOR lower = upper + 1 TO 5
diff = fnwt(lower) - fnwt(upper)
IF ABS(diff) < 1.5 THEN
PRINT #2, " = ";
ELSEIF diff > 0 THEN
PRINT #2, " ^ ";
ELSE PRINT #2, " v ";
END IF
NEXT
NEXT
FOR upper1 = 1 TO 4
FOR upper2 = upper1 + 1 TO 5
FOR lower1 = upper1 + 1 TO 4
IF lower1 <> upper2 THEN
FOR lower2 = lower1 + 1 TO 5
IF lower2 <> upper2 THEN
diff = fnwt(lower1) + fnwt(lower2) - fnwt(upper1) - fnwt(upper2)
IF ABS(diff) < 1.5 THEN
PRINT #2, " = ";
ELSEIF diff > 0 THEN
PRINT #2, " ^ ";
ELSE PRINT #2, " v ";
END IF
END IF
NEXT
END IF
NEXT
NEXT
NEXT
PRINT #2,
permute a$
LOOP UNTIL a$ = h$
CLOSE
where the subroutine permute is already shown elsewhere on the site.
To make sense of the table, I first tried an initial two weighings: a+b vs c+d and then a+c vs b+d. However this did not divide the possibilities very equally, and so I decided to proceed making the second weighing a+c vs b+e.
To simplify things, I assumed that I would base all the remaining weighings on the results of these first two weighings, rather than re-evaluate all successive weighings based on all the preceding weighings. As a result, my answer, 6 weighings may not be the best that can be done. But since even the theoretic best is 5 weighings, there is at most a waste of one weighing. The same table as above, but ordered by the results of the two weighings mentioned is:
a a a a b b b c c d ab ab ab ac ac ac ad ad ad ae ae ae bc bd be
b c d e c d e d e e cd ce de bd be de bc be ce bc bd cd de ce cd
abcde = ^ ^ ^ = ^ ^ = ^ = ^ ^ ^ ^ ^ ^ = ^ ^ = = = ^ ^ =
abced = ^ ^ ^ = ^ ^ ^ = = ^ ^ ^ ^ ^ ^ = = = = ^ ^ ^ = ^
abdce = ^ ^ ^ ^ = ^ = = ^ ^ ^ ^ = ^ ^ ^ ^ ^ = = = ^ ^ =
acbde ^ = ^ ^ = = ^ ^ ^ = ^ ^ ^ ^ ^ ^ = ^ ^ = = = ^ = v
acbed ^ = ^ ^ = ^ = ^ ^ = ^ ^ ^ ^ ^ ^ = = = = ^ ^ ^ v =
acebd ^ = ^ ^ v = = ^ = v ^ = ^ ^ ^ ^ = = = ^ ^ ^ ^ v =
adbce ^ ^ = ^ = = ^ v = ^ ^ ^ ^ = ^ ^ ^ ^ ^ = = = = ^ v
aecbd ^ ^ ^ = = = v ^ = v ^ = ^ ^ ^ ^ = = = ^ ^ ^ = v ^
a a a a b b b c c d ab ab ab ac ac ac ad ad ad ae ae ae bc bd be
b c d e c d e d e e cd ce de bd be de bc be ce bc bd cd de ce cd
badec = ^ = ^ ^ ^ ^ v = = ^ ^ ^ v v = = = ^ = v ^ = ^ ^
baecd = ^ ^ = ^ ^ ^ = = v ^ ^ ^ = v ^ v v = = = ^ ^ = ^
baedc = ^ ^ = ^ ^ ^ = v = ^ ^ ^ v v = = v ^ = = ^ = ^ ^
bcead v v = = = ^ ^ ^ = v ^ = ^ = v ^ v v v v = = ^ = ^
bdaec v ^ = = ^ = ^ v = ^ ^ ^ ^ v v v = = ^ = v = = ^ ^
bdeac v = v = ^ = ^ v v = ^ ^ = v v v = v ^ = v = = ^ ^
beacd v = ^ = ^ ^ = = v v ^ ^ ^ = v = v v v = = ^ ^ = ^
beadc v ^ = = ^ ^ = = v v ^ ^ ^ v v v = v = = = ^ = ^ ^
becad v = = v ^ ^ = ^ = v ^ = ^ = v = v v v v = ^ ^ = ^
bedac v = = v ^ ^ = v v = ^ ^ = v v v = v = = v ^ = ^ ^
eabcd = ^ ^ = = ^ v = v v ^ = = ^ v = = v v ^ ^ ^ = v ^
eabdc = ^ ^ = ^ = v = v v ^ = = = v v ^ v = ^ ^ ^ v = ^
eadbc ^ ^ = = = = v v v v ^ = v = v v ^ = = ^ ^ ^ v = ^
ebacd = = ^ v ^ ^ = = v v ^ = = = v = v v v ^ ^ ^ = v ^
ebadc = ^ = v ^ ^ = = v v ^ = = v v v = v = ^ ^ ^ v = ^
ebcad v = = v = ^ = ^ v v ^ v = = v = v v v = ^ ^ = v ^
ebdac v = = v ^ = = v v v ^ = v v v v = v = ^ = ^ v = ^
a a a a b b b c c d ab ab ab ac ac ac ad ad ad ae ae ae bc bd be
b c d e c d e d e e cd ce de bd be de bc be ce bc bd cd de ce cd
cadbe ^ = = ^ v = = ^ ^ ^ v = ^ ^ ^ ^ = ^ = v = v ^ = v
cadeb ^ = = ^ v v = ^ ^ = v v = ^ ^ ^ = ^ = = ^ v = v v
caedb ^ = ^ = v = v ^ ^ = v v = ^ ^ ^ = ^ v = ^ = = v v
cbdae v v = = = = ^ ^ ^ ^ v = ^ = ^ ^ v = = v v v ^ = v
cdabe = v = ^ v v = = ^ ^ v = = ^ ^ ^ = ^ = v v v ^ = v
cdaeb ^ v = = v v = = ^ ^ v v v ^ ^ ^ = ^ = = = v = v v
cdbae = v v = v = ^ = ^ ^ v = = = ^ ^ v ^ = v v v ^ = v
cdeab = v v = v v v = ^ = v v v ^ ^ = = ^ v = = v = v v
ceadb ^ v = = v = v ^ = v v v v ^ ^ ^ = = v = ^ = = v v
cedab = v = v v v v ^ = = v v v ^ ^ = = = v = ^ v = v v
dacbe ^ = = ^ = v = v ^ ^ v ^ = = ^ = ^ ^ ^ = v v = ^ v
daceb ^ = = ^ v v = v = ^ v = v = ^ = ^ ^ ^ ^ = v v = v
daecb ^ ^ = = = v v v = ^ v = v = ^ v ^ ^ ^ ^ = = v = v
dcabe = = v ^ v v = = ^ ^ v = = = ^ = ^ ^ ^ v v v = ^ v
dcaeb ^ = v = v v = = ^ ^ v v v = ^ = ^ ^ ^ = = v v = v
dcbae = v v = = v ^ = ^ ^ v = = v ^ = = ^ ^ v v v = ^ v
dceab = v v = v v v = = ^ v v v = ^ v ^ ^ = = = v v = v
a a a a b b b c c d ab ab ab ac ac ac ad ad ad ae ae ae bc bd be
b c d e c d e d e e cd ce de bd be de bc be ce bc bd cd de ce cd
dbcea v v v = = = ^ v = ^ v = v v v v = = = v v v = ^ v
dbeca v = v v ^ = = v = ^ v = v v v v = = = v v v v ^ =
debca v = v v = v = v v = v v v v v v = = = = v v v ^ =
decba = v v v = v v v = = v v v v v v = = = = v v v = v
ecbda v v = v = = v ^ = v v v v = v v v v v = = = = v ^
ecdba = v v v v = v = = v v v v = v v v v v = = = v v =
edbca v = v v = = v v v = v v v v v v = v v = = = v = ^
edcba = v v v = v v = v = v v v v v v = v v = = = v v =
a a a a b b b c c d ab ab ab ac ac ac ad ad ad ae ae ae bc bd be
b c d e c d e d e e cd ce de bd be de bc be ce bc bd cd de ce cd
bcdae v v = = = ^ ^ = ^ ^ = ^ ^ v = ^ v = = v v v ^ ^ =
bdcae v = v = ^ = ^ = ^ ^ = ^ ^ v = = v = ^ v v v ^ ^ =
cbead v v = = = ^ = ^ ^ v = v ^ ^ = ^ v v v v = = ^ v =
cebad = v = v v ^ = ^ = v = v = ^ = ^ v v v v ^ = ^ v =
dabec = ^ = ^ ^ v = v = ^ = ^ = v = v ^ ^ ^ ^ v = v ^ =
daebc ^ ^ = = = v = v v ^ = ^ v v = v ^ ^ ^ ^ = = v ^ =
dbace = = v ^ ^ = ^ v = ^ = ^ = v = = ^ ^ ^ v v v = ^ v
eacdb ^ = ^ = v = v = v v = v v ^ = = ^ = v ^ ^ ^ v v =
eadcb ^ ^ = = = v v = v v = v v ^ = v ^ = = ^ ^ ^ v v =
ecabd = = ^ v v = v ^ = v = v = ^ = = v v v ^ ^ ^ = v ^
a a a a b b b c c d ab ab ab ac ac ac ad ad ad ae ae ae bc bd be
b c d e c d e d e e cd ce de bd be de bc be ce bc bd cd de ce cd
bcdea v v v = = ^ ^ = ^ = = = = v v = v v v v v v ^ ^ =
bceda v v = v = ^ ^ ^ = = = = = v v = v v v v v v ^ = ^
bdcea v v v = ^ = ^ = = ^ = = = v v v v v = v v v ^ ^ =
bdeca v = v v ^ = ^ v = = = = = v v v v v = v v v = ^ ^
becda v v = v ^ ^ = = = v = = = v v v v v v v v = ^ = ^
bedca v = v v ^ ^ = = v = = = = v v v v v v v v = = ^ ^
dbaec = ^ v = ^ = ^ v = ^ = ^ = v v v ^ ^ ^ = v = v ^ =
dbeac v = v = ^ = = v v ^ = ^ v v v v ^ = ^ = v = v ^ =
deabc = ^ v = = v v v v = = = v v v v ^ ^ ^ ^ = = v ^ =
debac = = v v ^ v = v v = = = v v v v ^ = ^ ^ v = v ^ =
ebcda v v = v = ^ = = v v = v v v v v v v v = = = = v ^
ebdca v = v v ^ = = = v v = v v v v v v v v = = = v = ^
ecbad = v = v = ^ v ^ = v = v = ^ v = v v v = ^ ^ = v ^
edabc = ^ = v = v v v v = = = v v v v ^ = = ^ ^ ^ v = ^
edbac = = v v ^ = v v v = = = v v v v ^ v = ^ = ^ v = ^
a a a a b b b c c d ab ab ab ac ac ac ad ad ad ae ae ae bc bd be
b c d e c d e d e e cd ce de bd be de bc be ce bc bd cd de ce cd
cbdea v v v = = = ^ ^ ^ = v v = = = = v v v v v v ^ = v
cbeda v v = v = ^ = ^ ^ = v v = = = = v v v v v v ^ v =
cdbea v v v = v = = = ^ ^ v v v = = = v = v v v v ^ = v
cdeba = v v v v v = = ^ = v v v = = = v = v v v v = v v
cebda v v = v v = = ^ = v v v v = = = v v v v = v ^ v =
cedba = v v v v = v ^ = = v v v = = = v v v v = v = v v
dbcae v = v = = = ^ v ^ ^ v ^ = v = = = ^ ^ v v v = ^ v
dcbea v v v = = v = = ^ ^ v v v v = v = = = v v v = ^ v
dceba = v v v v v = = = ^ v v v v = v = = = v v v v = v
deacb ^ = v = = v v v v = v v v = = v ^ ^ ^ ^ = = v = v
decab = = v v v v v v = = v v v = = v ^ ^ = ^ = v v = v
ecadb ^ = = v v = v ^ = v v v v ^ = = = = v ^ ^ ^ v v =
ecdab = v = v v v v = = v v v v ^ = v = = v ^ ^ = v v =
edacb ^ = = v = v v v v = v v v = = v ^ = = ^ ^ ^ v v =
edcab = = v v v v v = v = v v v = = v ^ = v ^ ^ = v v =
a a a a b b b c c d ab ab ab ac ac ac ad ad ad ae ae ae bc bd be
b c d e c d e d e e cd ce de bd be de bc be ce bc bd cd de ce cd
acdbe ^ = ^ ^ v = = = ^ ^ = ^ ^ ^ ^ ^ ^ ^ ^ = = = ^ = v
acdeb ^ = ^ ^ v v = = ^ = = = = ^ ^ ^ ^ ^ ^ ^ ^ = = v v
acedb ^ = ^ ^ v = v ^ = = = = = ^ ^ ^ ^ ^ = ^ ^ ^ = v v
adcbe ^ ^ = ^ = v = = ^ ^ = ^ ^ ^ ^ ^ ^ ^ ^ = = = = ^ v
adceb ^ ^ = ^ v v = = = ^ = = = ^ ^ ^ ^ ^ ^ ^ ^ = v = v
adecb ^ ^ = ^ = v v v = = = = = ^ ^ = ^ ^ ^ ^ ^ ^ v = v
aecdb ^ ^ ^ = v = v = = v = = = ^ ^ ^ ^ ^ = ^ ^ ^ v v =
aedcb ^ ^ ^ = = v v = v = = = = ^ ^ = ^ ^ ^ ^ ^ ^ v v =
cabde = = ^ ^ v = ^ ^ ^ = = = ^ ^ ^ ^ v ^ = v = v ^ = v
cabed = = ^ ^ v ^ = ^ ^ = = = ^ ^ ^ ^ v = v v ^ = ^ v =
caebd ^ = ^ = v = = ^ ^ v = v ^ ^ ^ ^ v = v = ^ = ^ v =
cbade = v = ^ = ^ ^ ^ ^ = = = ^ ^ ^ ^ v = = v v v ^ = v
cbaed = v ^ = = ^ ^ ^ ^ = = = ^ ^ ^ ^ v v v v = = ^ v =
ceabd = v ^ = v = v ^ = v = v = ^ ^ ^ v v v = ^ = ^ v =
dabce = ^ = ^ = v ^ v = ^ = ^ = v ^ = ^ ^ ^ = v v = ^ v
a a a a b b b c c d ab ab ab ac ac ac ad ad ad ae ae ae bc bd be
b c d e c d e d e e cd ce de bd be de bc be ce bc bd cd de ce cd
abdec = ^ ^ ^ ^ = ^ v = = ^ ^ ^ = = = ^ ^ ^ ^ = ^ = ^ ^
abecd = ^ ^ ^ ^ ^ = = = v ^ ^ ^ ^ = ^ = = = ^ ^ ^ ^ = ^
abedc = ^ ^ ^ ^ ^ = = v = ^ ^ ^ = = = ^ = ^ ^ ^ ^ = ^ ^
adbec ^ ^ = ^ ^ = = v = ^ ^ ^ ^ = = = ^ ^ ^ ^ = ^ v ^ =
adebc ^ ^ = ^ = v = v v = ^ ^ = = = = ^ ^ ^ ^ ^ ^ v ^ =
aebcd ^ ^ ^ = = ^ = = v v ^ ^ ^ ^ = ^ = = = ^ ^ ^ = v ^
aebdc ^ ^ ^ = ^ = = = v v ^ ^ ^ = = = ^ = ^ ^ ^ ^ v = ^
aedbc ^ ^ ^ = = = v v v = ^ ^ = = = = ^ ^ ^ ^ ^ ^ v = ^
bacde = = ^ ^ ^ ^ ^ = ^ = ^ ^ ^ = = ^ v = ^ v v = ^ ^ =
baced = = ^ ^ ^ ^ ^ ^ = = ^ ^ ^ = = ^ v v = v = ^ ^ = ^
badce = ^ = ^ ^ ^ ^ = = ^ ^ ^ ^ v = ^ = = ^ v v = ^ ^ =
bcade v = = ^ = ^ ^ ^ ^ = ^ ^ ^ = = ^ v = = v v v ^ ^ =
bcaed v = ^ = = ^ ^ ^ ^ = ^ ^ ^ = = ^ v v v v = = ^ = ^
bdace v = = ^ ^ = ^ v = ^ ^ ^ ^ v = = = = ^ v v v ^ ^ =
eacbd ^ = ^ = = = v ^ v v ^ v = ^ = = = v v ^ ^ ^ = v ^
a a a a b b b c c d ab ab ab ac ac ac ad ad ad ae ae ae bc bd be
b c d e c d e d e e cd ce de bd be de bc be ce bc bd cd de ce cd
Depending on which of the nine possible results derive from the first two weighings, the remaining four weighings are shown in the following table, in which the weighings not used are left blank:
grams
55555 a a a a b b b c c d ab ab ab ac ac ac ad ad ad ae ae ae bc bd be
12345 b c d e c d e d e e cd ce de bd be de bc be ce bc bd cd de ce cd
abcde = ^ = ^ ^ ^
abced = ^ = ^ ^ =
abdce = ^ ^ ^ ^ ^
acbde ^ = = ^ ^ ^
acbed ^ = = ^ ^ =
acebd ^ = v ^ ^ =
adbce ^ ^ = ^ ^ ^
aecbd ^ ^ = ^ ^ =
a a a a b b b c c d ab ab ab ac ac ac ad ad ad ae ae ae bc bd be
b c d e c d e d e e cd ce de bd be de bc be ce bc bd cd de ce cd
badec = ^ ^ v ^ v
baecd ^ = ^ v = =
baedc ^ = ^ v ^ =
bcead = = ^ v v =
bdaec = = ^ v ^ v
bdeac v = ^ v ^ v
beacd ^ = ^ v v =
beadc = = ^ v = =
becad = v ^ v v =
bedac = v ^ v = v
eabcd ^ = ^ v v ^
eabdc ^ = ^ v = ^
eadbc = = &n
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Posted by Charlie
on 2004-08-02 16:08:36 |
Don't know if this is the fewest weighings...
