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 Five Weights and a Broken Scale (Posted on 2004-08-02)
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.

 See The Solution Submitted by Brian Smith Rating: 3.5556 (9 votes)

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 The rest of my comment | Comment 2 of 17 |
(In reply to Don't know if this is the fewest weighings... by Charlie)

That one got cut off.  From the beginning of the bottom table:

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:

`grams55555 a  a  a  a  b  b  b  c  c  d  ab ab ab ac ac ac ad ad ad ae ae ae bc bd be12345 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 cdbadec       =  ^                    ^           v           ^     vbaecd       ^  =                    ^           v           =     =baedc       ^  =                    ^           v           ^     =bcead       =  =                    ^           v           v     =bdaec       =  =                    ^           v           ^     vbdeac       v  =                    ^           v           ^     vbeacd       ^  =                    ^           v           v     =beadc       =  =                    ^           v           =     =becad       =  v                    ^           v           v     =bedac       =  v                    ^           v           =     veabcd       ^  =                    ^           v           v     ^eabdc       ^  =                    ^           v           =     ^eadbc       =  =                    ^           v           =     ^ebacd       ^  v                    ^           v           v     ^ebadc       =  v                    ^           v           =     ^ebcad       =  v                    ^           v           v     ^ebdac       =  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 cdcadbe       =  ^                    v           ^           =     =cadeb       =  ^                    v           ^           =     ^caedb       ^  =                    v           ^           v     ^cbdae       =  =                    v           ^           =     vcdabe       =  ^                    v           ^           =     vcdaeb       =  =                    v           ^           =     =cdbae       v  =                    v           ^           =     vcdeab       v  =                    v           ^           v     =ceadb       =  =                    v           ^           v     ^cedab       =  v                    v           ^           v     ^dacbe       =  ^                    v           ^           ^     vdaceb       =  ^                    v           ^           ^     =daecb       =  =                    v           ^           ^     =dcabe       v  ^                    v           ^           ^     vdcaeb       v  =                    v           ^           ^     =dcbae       v  =                    v           ^           ^     vdceab       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 cddbcea v  v        =                 v           v        =dbeca v  =        ^                 v           v        =debca v  =        =                 v           v        =decba =  v        =                 v           v        =ecbda v  v        =                 v           v        vecdba =  v        v                 v           v        vedbca v  =        =                 v           v        vedcba =  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 cdbcdae v  v        =                 =           =        =bdcae v  =        ^                 =           =        =cbead v  v        =                 =           =        vcebad =  v        v                 =           =        vdabec =  ^        ^                 =           =        ^daebc ^  ^        =                 =           =        ^dbace =  =        ^                 =           =        ^eacdb ^  =        v                 =           =        =eadcb ^  ^        =                 =           =        =ecabd =  =        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 cdbcdea    v           ^  ^           =           v                          ^bceda    v           ^  ^           =           v                          =bdcea    v           =  ^           =           v                          ^bdeca    =           =  ^           =           v                          ^becda    v           ^  =           =           v                          =bedca    =           ^  =           =           v                          ^dbaec    ^           =  ^           =           v                          ^dbeac    =           =  =           =           v                          ^deabc    ^           v  v           =           v                          ^debac    =           v  =           =           v                          ^ebcda    v           ^  =           =           v                          vebdca    =           =  =           =           v                          =ecbad    v           ^  v           =           v                          vedabc    ^           v  v           =           v                          =edbac    =           =  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 cdcbdea v                    ^  ^     v           =                          =cbeda v                    ^  ^     v           =                          vcdbea v                    =  ^     v           =                          =cdeba =                    =  ^     v           =                          vcebda v                    ^  =     v           =                          vcedba =                    ^  =     v           =                          vdbcae v                    v  ^     v           =                          ^dcbea v                    =  ^     v           =                          ^dceba =                    =  =     v           =                          =deacb ^                    v  v     v           =                          =decab =                    v  =     v           =                          =ecadb ^                    ^  =     v           =                          vecdab =                    =  =     v           =                          vedacb ^                    v  v     v           =                          vedcab =                    =  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 cdacdbe    =           =  =           =           ^                          =acdeb    =           v  =           =           ^                          vacedb    =           =  v           =           ^                          vadcbe    ^           v  =           =           ^                          ^adceb    ^           v  =           =           ^                          =adecb    ^           v  v           =           ^                          =aecdb    ^           =  v           =           ^                          vaedcb    ^           v  v           =           ^                          vcabde    =           =  ^           =           ^                          =cabed    =           ^  =           =           ^                          vcaebd    =           =  =           =           ^                          vcbade    v           ^  ^           =           ^                          =cbaed    v           ^  ^           =           ^                          vceabd    v           =  v           =           ^                          vdabce    ^           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 cdabdec =                    v  =     ^           =                          ^abecd =                    =  =     ^           =                          =abedc =                    =  v     ^           =                          ^adbec ^                    v  =     ^           =                          ^adebc ^                    v  v     ^           =                          ^aebcd ^                    =  v     ^           =                          vaebdc ^                    =  v     ^           =                          =aedbc ^                    v  v     ^           =                          =bacde =                    =  ^     ^           =                          ^baced =                    ^  =     ^           =                          =badce =                    =  =     ^           =                          ^bcade v                    ^  ^     ^           =                          ^bcaed v                    ^  ^     ^           =                          =bdace v                    v  =     ^           =                          ^eacbd ^                    ^  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`

or in concise terms, if the first two weighings both showed the same result (^^ or == or vv), then make the subsequent weighings a vs b, a vs c, b vs c and a+d vs b+e.

If however the first two weighings were ^v or v^, then the next four weighings would be a vs d, a vs e, a+d vs c+e and a+e vs b+d.

If the first two weighings had been =v or =^, the next four would have been a vs c, b vs d, b vs e and b+d vs c+e.

And if the first two had been v= or ^=, the next four would have been a vs b, c vs d, c vs e and b+d vs c+e.

This shows how to order the weights in 6 weighings.  In some instances above, the sixth weighing (that is, one of the extra four) may have been unnecessary, but six are shown in all instances.  It may be possible to do it in five weighings, but no less, based on information theory.

 Posted by Charlie on 2004-08-02 16:12:29

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