I figured out the solution to the puzzler (for some reason I only get the puzzler e-mail every other week, so I’m a bit behind here), but the solution posted is seriously flawed. They state “Clearly, one of the stone pieces has to be one pound” but they don’t give a reason. I started out assuming there was a one pound stone to weigh the one pound amount, but quickly realized that two stones that differed by one pound would also work.
So I turned around and started from the other end of the scale: 40, 39, 38, etc. The only way to measure 39 lbs is if one stone (left off the scale) was 1 lb and 38 lb was 39-1. The most logical way to measure 37 lb was if there was 3 lb stone left off. The only other option was both a 1 lb and a 2 lb stone and it quickly became apparent that there was no way to weight all 40 amounts with 4 pieces. After I had 1 & 3, I guessed the next higher weight was 7 but quickly realized (working from the low end of the weights, 1, 2, 3, etc) that with a 9 lb piece I would be able to measure 5-13 lbs, and that left a 27 lb piece which I checked could make the rest of the measurements. But it isn’t until I had that 3rd stone (9 lb) that the progression 1-3-9-27 was apparent, and at that time the 4th stone’s weight just falls out. But I’m sure that very few people who get 1-3 will figure out "oh, these are powers of 3, the next number must be 9, and, ah ha, that makes the last stone 27, so that has to be the solution. Maybe if there were 5 stones weighing 121 lb, and someone figured out 1-3-9 they might see the pattern and go immediately to 27, but with only 1-3 the pattern really isn’t apparent.
The hint “how would you weigh two pounds?” was worthless. First, they didn’t justify why a stone had to be one lb, then they jumped to the next stone having to be 3 lb since 3-1=2, but any two stones that differ by 2 lb could be used for the 2 lb measurement.
ssather — "Maybe if there were 5 stones weighing 121 lb, and someone figured out 1-3-9 they might see the pattern and go immediately to 27, but with only 1-3 the pattern really isn't apparent."
"At first you are tempted to run up the powers of 2. Weights of 1, 2, 4, and 8 pounds, in various combinations, can weigh everything from 1 to 15 pounds, but that’s nowhere near 40. The trick is, some of the weights can sit on the left pan, with the box. Each weight can be on the left, the right, or not used at all. This suggests powers of 3.
Using 1 and 3 pound weights, you see immediately how to measure 1, 3, and 4 pounds. If the box weighs 2 pounds, put the 1 pound weight on the left pan with the box, and the 3 pound weight on the right, and everything is in balance. Next, bring in a 9 pound weight and put it on the right. We already showed we can add a net weight of 1 through 4 to the box, so if the box weighs anywhere from 5 to 8 pounds, we’ll be able to balance it against the 9 pound weight. Of course there is no trouble if the box weighs 9 pounds, and if it weighs anywhere from 10 to 13 pounds, we achieve balance by adding (net) 1 to 4 pounds to the right. Now we’ve covered everything from 1 to 13 pounds, and the last weight to bring in is 27 pounds. This lets us weigh everything from 27-13 pounds up to 27+13 pounds, or 40 pounds. An inductive proof shows successive weights should always be powers of 3.
To weigh a box from 1 to 81 pounds, double the weights. Thus the weights are 2, 6, 18, and 54 pounds. If the weight of the box is even, you can determine it using the procedure outlined above. If the weight is odd, the balance will always tip one way or the other. Determine that the box is heavier than n and lighter than n+2. For instance, the box might be more than 30 pounds and less than 32 pounds, whence it is 31 pounds. http://www.eklhad.net/funmath.html
rudy</b? — — "today's puzzler is so obfuscatedly worded that it makes no sense: If there are 20,000 lights, at some point someone is going to come skipping along and pull every 20,000th chain. What is meant by "every 20,000th chain?" I thought there were only 20,000 lights total. So how can there be more than 20,000 chains?"
20,000 is the Red Sox Nation's concept of infinity. But I give you points for using the word obfuscatedly (although the word you really wanted was obfuscatorily).
It comes down to a question of whether the “light number” has an even or odd number of factors. For example, consider hall light number “8.” The number 8 has an even number of factors: 1, 2, 4, and 8. Hence persons 1, 2, 4, & 8 yank the pull chain — an even number of yanks leaves the light bulb in its original state: OUT. Next consider hall light number “9.” The number 9 has an odd number of factors: 1, 3, & 9. An odd number of chain yanks leaves the light bulb in a changed state: ON.
What is the difference between the integers 8 and 9? MIT graduates? Bostonians? Anyone? If you heard this Puzzler on the radio in downtown Chicago, you could have solved it before the next stop-light.
I should submit some University of Chicago Lab School entrance exams (grades K-8) for Click & Clack to mull over.
(My ex lives in Needham, so you can see where I’m going.)
Rude_man: Look on the right-hand side of the screen, under the cartoony-looking logo for the Car Talk Community. There’s a rectangular red button that says, “Ask a Question.” When you click on it, it brings up a form to submit your query.
Referring to the Stone Temple Farmers puzzle.
I really enjoyed this puzzle. The hint about 2 was good because if you guess that the lightest weight is 1 (really the simplest assumption) than the next guess for the second heaviest weight is 3. Guessing some more, you get the answer. I should add that I just heard of this puzzle today because the show is being rebroadcast.
I liked the 4 stone puzzler too. I brute forced it, then later came up w/the following logic, which made it simpler.
The mathematical objective is to – as efficiently as possible – use a combination of 4 numbers (stones) using either addition or subtraction to make any number from 1 to 40. (If subtraction isn’t allowed, only addition, then you’re forced to use the common binary weights 2^0, 2^1, 2^2 … .)
Makes common sense to start with 1 as the first number.
The next number has to be 3. Why? 1&2 combined can only make a number from -3 to +3; but 1 & 3 can be combined to make any number from -4 to + 4. So to be most efficient, the next number has to be 3. 1 & 4 would miss 2, so 4 couldn’t be the next number.
So how to get from 4 to 5 then? Since you already have the ability to make 4, the next number (stone) has to be 5+4, or 9. Anything less than 9 would be redundant, and more than 9 would leave 5 (and possibly others) not reachable. With 1,3, & 9 you can now make any number from -13 to + 13.
Using the same logic, the final stone has to be 14 + 13 = 27.