Daring stumbled out of the stone door, coughing. The Room of Fire was pretty good as far as traps were concerned: it had almost entirely singed her jacket and hat and her face was completely covered in ash, but she was alive for the most part. The Trial of the Body was not one to be overcome easily.
The pegasus slowly made her way towards the next stone door at the other end of the corridor. There she was greeted by what appeared to a sigil, with an outline where a hoof was apparently meant to go, judging by its shape. Gingerly, Daring raised her left hoof to the mark, and tapped it. Instantly, a purple light erupted from the floor, and a strange, ethereal shape began to form to the right of Daring Do, causing her to sharply back away in alarm. After about ten seconds, the purple glow began to fade, leaving behind a semi-transparent outline of what appeared to be a mountain goat, with fur as long as the lies of her enemy Ahuizotl, with a matching beard reaching all the way down to the floor and trailing for about three feet.
"I am the Ancient Goat," the spirit said. "You who seek the treasures within, you have escaped the room of fire so you are surely brave. But let us see if your wits are just as abundant. Let me be your test!"
Daring nodded, regaining her senses. No archeological adventure had any puzzle that she couldn't solve, or any trap that she couldn't avoid, so she was feeling pretty confident. The moment she accepted, a purple smoke appeared right under Daring's feet. She quickly stepped back, and instantly, a set of old fashioned weighing balances materialised in front of Daring, along with what appeared to be twelve bronze coins in one of its pans.
Daring's face grimaced. To be honest, she had recognised the puzzle on sight as soon as it appeared, and she had to confess herself disappointed. "I have here," the Ancient Goat intoned, "a set of twelve identical looking coins and-"
"Really?" Daring stared at the offending items in front of her. "The coins and scales puzzle? Here I was hoping an ancient and influential civilization would have a shred of originality..."
"It was a different time!" protested the Ancient Goat. "We had fewer logicians back then!"
Daring rolled her eyes. "Okay, fine. So I have to find the one coin that's different, right? Is the coin heavier or lighter than the others?"
"That is part of the puzzle," the Ancient Goat smiled. "You need to find that out yourself."
Daring paused. "Okay, credit where it's due, that's a bit trickier. And how many weighings am I allowed?"
"Three," the Ancient Goat replied. "I do advise you give it a good think, I should say. Take the wrong coin, and you will find all respiratory privileges...rescinded from the curse of my ancestors. Pass my test, and you will be allowed forth, to the Trial of the Heart!"
Daring nodded. Despite how thoroughly outdated the puzzle was as far as modern society was concerned, a puzzle was a puzzle, and she couldn't wait for Ahuizotl to start catching up to her. "Right, better get cracking then!"
Well this is easier then the last daring do puzzle
weigh 4 coins on each end
If one is heavier it has the coin, if not its one of the 4 you didn't weigh
Now we have 4 coins, weigh 1 on each side, if it tips we're done if not try the other 2 coins then it must tip
Or
weigh 3 on each side, if it tips it one of the 3, if not its one of the other 6 so test them
Now we have 3 coins and either 1or2 weighings left, weigh 1 coin on each end, if it tips you got it if not its the last coin of the 3
Im surprised i could find another solution
EDIT: ok i think i found a 3rd option
weigh 6 on each, it must tip
Weigh 3 of the heavier side of 6 on each, must tip again
Weigh one of the 3 on eacj, if it tips you got it, if not its the coin of the 3 you didn't just weigh
EDIT 2:ok how many solutions ARE THERE?
weigh 6 on each, take heavy side
Weigh 2 of the 6 on each, if it tips it one of the 2, if not its one of the 2 not weighed
At 2 coins its a simple last check to see which one is the odd coin
9202345
Could take place before then
9202367
WOW, i did 4 trys only to now realize we dont know if the coin in question is lighter or heavier, man am i dumb
This changes the plan significantly
weight coins 1-4 vs 5-8
If that balances odd coin is between 9-12
Then weight 3 good coins(ie any we know are equal) vs 3 unknown coins, if its even then 4th unknown coin is odd, we can spend the third weigh on checking it with any good coin to see how its different
If the second weighing is off take note if the 3 unknown coins go up or down, if its up one of 3 unknown is lighter, if down one is heavier, weigh 2 of them and see how it tips,using the previous weighing to determine what your looking for, if they tip then there we go, if not the 3rd coin is what we want
Now the tricky part is if the first weighing is off
If that happens then take note witch side of 4 coins is heavy (in my case ill say 5-8 is)
Weigh coins 1,5&6 vs 2,7&8, if this is equal then either 3 or 4 is odd, weight 4 vs 9(in this case 9 is good) if equal 3 is odd, if not 4 is odd and based on the tilts its heavy or light
If the 1,5&6 vs 2,7&8 is off then note the heavy side
If 2,7&8 is heavy weigh 7&8,if equal 1 is light,if not 7 or 8 is heavy
If 1,5&6 is heavy weigh 5&6, if equal 2 is light, if not 5 or 6 is heavy
Love how you say well known, and i have never seen this puzzle b4
9202457
Every variation I've seen of this states the weight of the fake. It's got a TV Tropes article, so it must be pretty famous.
OH NO NOT THIS ONE
9204491
It gave me no pleasure either but I needed to churn one out for the day and it was there
You can even do this with 13 balls, you know? (26 possibilities < 3x3x3, I worked out the solution once) But I'll provide the solution for 11.
abc vs def
case =: we know it's in ghij -> ghi vs abc
case = then = : it's j, -> a vs j
case = then = then >: j is heavy
case = then = then <: j is light
case = then = then =: you screwed up
case = then <>: g vs h.
case = then > then <>: heavier one is heavy
case = then < then <>: lighter one is light
case = then > then =: i is heavy
case = then < then =: i is light
cases <>: relabel them as needed to make abc heavier than def. Then weigh ad vs be.
case <> then =: we know it's c heavy or f light. Weigh c vs g.
case <> then = then =: f is light
case <> then = then >: c is heavy
case <> then = then <: you screwed up
case <> then >: a is heavy or d is light. do as above with a taking the role of c and d taking the role of f
case <> then <: b is heavy or e is light. do as above with b taking the role of c and e taking the role of f
Step 1: let us take 8 coins randomly. Put 4 of them on the left pan and 4 on the right pan. If they are all identical, then the one we seek is among the 4 yet unweighed, proceed to step 2.1. If not, then that we seek is on one of the pans, proceed to step 2.2.
Step 2.1: clear one of the pans and take a half of coins off the other pan, then select two random coins from the pile yet unweighed and put them on the cleared pan. If these 4 coins are still balanced, then the two yet unweighed must be weighed, proceed to step 3.1. If not, the two random coins we selected are to be distinguished somehow, proceed to step 3.2.
Step 3.1: as in step 2.1, clear one of the pans and take a half of the coins off the other pan, then select a random coin from the two yet unweighed and put it on the cleared pan. If they are still balanced, then that we did not select in this step must be the different coin. If not, then the coin we did select is the different one. It has been found in three steps, as was required.
Step 3.2: by this point we have two coins, which we know to be identical, on one pan and two other coins, one of which is different, on the other pan. We take one coin off the pan with the identical coins and toss it away, then take one random coin off the other pan and hold it tightly. If the scales are balanced now, then the one we are holding is the different one. If not, then the one that remained on the pan, where we had suspected the target coin to be, must be the different one. Again been found in three steps.
Step 2.2: by this point we have 4 coins on one pan and 4 on the other pan, and the different one is among these 8 coins. Before we take measurement, we take a half of the coins off both pans, so that only 2 vs 2 remain, and also take the two coins we know to be identical from the prev. step and replace one of the pairs being weighed by them (only now do we perform the second weighing!). Call the pairs removed before the replacement A and B, then the pair after the replacement be called R ("replaced"). If the scales are not balanced, then we know, on which pan it is (since one of the pans contains the coins we know to be identical), so remove them, name them "yet unweighed" and proceed to step 3.1. However, if the scales are balanced, then the one we seek is either in A, or in B, or in R. And that's a problem, so step 2.2 must be changed...
I'll think of something later, when I have slept...
OK, methinks, I have figured this out.
As was stated in my previous comment about a week ago, we take 8 coins out of 12 and weigh them. If the coins are balanced, then the target coin is among the 4 yet unweighed, so, with the reference weight of the 8 we weighed, we weigh the 4 and find the two possible candidates, and then with the third weighing we determine the target coin. However if the 8 coins are unbalanced, then we must determine the target coin in yet three weighings (because it's binary search, duh), and we only have yet two...
But Daring Do would not have been Daring Do, if she had had no wits! With a cool expression of the mare as cool, as Rainbow Dash ), she makes the two remaining weighings and is left with two candidates -- no more weighings allowed. She must now answer, which one is the different. It would be much easier, were just one measurement allowed to her, but, oh, well, decides she and... gets rid of one of the coins! The Ancient Goat is... surprised... to say the least. Unwavered, Daring Do tells the Goat with a smile, that this coin left is the different one! It MUST be the one! All of her measurements showed, that all of the previous coins had been identical, and only this is left, so it MUST be the one! And I don't think the Ancient Goat could afford weighing the coin one more time. If he knew, which coin is different, he would have to prove first, that it IS. Can't argue with logic (unless you are Pinkie... or Discord)!
Another solution may be, I think, Daring Do could have weighed 6 out of 6, thus finding the reference weight, then narrowed her results to a grouppe of 3 coins, and then gotten rid of one and THEN weighed one of the TWO remaining (that would have been more convincing ).