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A weighing problem



Nine identical looking cubes but two off by 1 gram (the two not necessarily equal to each other). However the cubes have a particular trait, when numbered they magically gain the weight of the number written on them. Without numbering there is no physical way to distinguish one cube from the next. You were hired by the owner of the nine cubes to number the cubes in such a way that the two cubes different from the others can be identified easily.

To assist with finding the cubes you have a balance scale and a marker. What is the least number of weightings needed to find the off cubes and what will be your numbering system?

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I'll define a unit of weight called a "gerbleflutz" as the weight of a normal cube, and I'll write negative one gerbleflutz on each of the cubes. I then lift each cube into the air and hold it very still while letting go. Those that float to the ceiling or fall to the floor are the odd ones out, and the scale needn't be used at all.

I then let someone else try a more serious shot at the problem, because this one's tough.

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sub solution
for 1st & 2nd trials are balanced:

Number system:
3 2
4 5
6 7
9 8

1st try 3469 v 2578

2nd if =, try 29 47 v 38 56
if = ,suspect 36,49,27,58 are (+1 -1) pairs
23,26,37,45,48,59,67,89 are (+1 ) or (-1 ) pairs

3rd try 348 v 267
if = , 27 is (+1 -1) pair
59 is (+1 ) or (-1 ) pair

try 278 v 359
if =, 27
if<>, 59

if > ,suspect 48 (+1)pair
or 26,67 (-1 )pairs

try 597 v (16)32
if = , 48
if <>, 6 & lighter (7 or 2)

if < ,suspect 48 (-1 )pair
or 26,67 (+1)pairs

try 597 v (16)32
if = , 48
if <>, 6 & heavier (7 or 2)

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