Brain Teasers
Marbles
You are filling various lengths of tubes with marbles. A yellow marble is 19 mm wide and a green marble is 21 mm wide. How many marbles of each color will exactly fill a tube that is 562 mm long?
Answer
13 yellow and 15 green marbles.13*19=247
15*21=315
247+315=562
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Comments
Jan 16, 2006
I disagree. The number of marbles that will fit in this tube is entirely dependent on the diameter of the tube. As written, the tube must be at least 21mm in diameter to accomodate even 1 of the 21mm marbles. However, this will cause the 19mm marbles to sit unevenly, so the length from the end of one marble to the end of the second will not be simply the sum of the diameters, but rather, you need to take into accouunt how far the marbles sit to the side.
However, if the diameter is much larger than the diameter of either marble, who knows how many you could put in?
However, if the diameter is much larger than the diameter of either marble, who knows how many you could put in?
i agree anmartin i did not read the ans. yet, how could you determine the ans. if no dia. is given.
OK, you guys are technically correct, but the spirit of the question was "how many of each marble, when perfectly aligned edge to edge, will total a length exactly equal to 562 mm?"
Or suppose the tube was 21mm diameter with a 1 mm spongy inner lining that kept all the marbles perfectly centered in the tube. That would settle the ambiguity.
Or suppose the tube was 21mm diameter with a 1 mm spongy inner lining that kept all the marbles perfectly centered in the tube. That would settle the ambiguity.
they're alright a as what they said...how come?
but....its ok...great teaser...
but....its ok...great teaser...
What's the point of working on a teaser if it's not technically accurate? It was apparent what was intended in the question, but it also was not correctly answered.
Do you have a method for solving this teaser other than trial and error?
if you take a yellow marble and a green marble, you would get 40mm. Do this 14 times, making 14 marbles of each color, you would have 560mm. Change one of the yellow marbles to a green marble to add 2mm of length to make 562mm, thereby making the solution of 13 yellow marbles and 15 green.
The formula is:
19x + 21y = 562
19x = 562 - 21y
x = (562 - 21y) / 19
excessively (for those of you not using spreadsheets)
x = 29 + ((11 - 21y) / 19)
x = 28 + ((30 - 21y) / 19)
.
.
x = 13 + ((315 - 21y) / 19)
A look at some modular math will help you to project and skip most fruitless trials.
19x + 21y = 562
19x = 562 - 21y
x = (562 - 21y) / 19
excessively (for those of you not using spreadsheets)
x = 29 + ((11 - 21y) / 19)
x = 28 + ((30 - 21y) / 19)
.
.
x = 13 + ((315 - 21y) / 19)
A look at some modular math will help you to project and skip most fruitless trials.
Bye the bye-
Stringing beads on a wire would be a satisfactory way to avoid tube diameter dilemma. I wouldn't mind have to picture an infinitely thin wire.
Stringing beads on a wire would be a satisfactory way to avoid tube diameter dilemma. I wouldn't mind have to picture an infinitely thin wire.
The problem easily solves without trial and error.
562 / 21 = 26, remainder 16. Add 21 to make at least 562, you now have 27 21-mm marbles and 5 mm extra. Each 21-mm marble you replace with a 19-mm marble will reduce the length by 2, so the difference needs to be even. Add another 21-mm marbles to get 28 21-mm marbles and 26 extra. You now need to replace 26/2 = 13 21-mm marbles with 19-mm marbles to take care of the 26 extra. 28 - 13 = 15 21-mm marbles.
562 / 21 = 26, remainder 16. Add 21 to make at least 562, you now have 27 21-mm marbles and 5 mm extra. Each 21-mm marble you replace with a 19-mm marble will reduce the length by 2, so the difference needs to be even. Add another 21-mm marbles to get 28 21-mm marbles and 26 extra. You now need to replace 26/2 = 13 21-mm marbles with 19-mm marbles to take care of the 26 extra. 28 - 13 = 15 21-mm marbles.
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