For the "zero is possible" case: On the first evening if Rose has 0 bars and is not free, she knows that Mark does not have 20 bars and hence may deduce there are 18 in total. Similarly if she has 1 bar and is not free she knows that Mark does not have 19 and again may deduce there are 18 in total. If she has 19 or 20 she can say 20, just as Mark could in the morning. On the second morning if Mark has 18 bars and is not free, he knows that Rose does not have 0 bars and hence may deduce there are 20 in total.
Similarly if he has If he has 0 bars and is not free he knows Rose does not have 20 and hence may deduce there are 18 in total. Similarly if he has 1. For the "zero is not possible" case, the sets of values get shifted away from 20 and 0 by one, because they have the knowledge that the other window does not have zero bars. I have an alternative answer which, if correct, would allow Mark to give an answer on the morning of the second day.
I will be asked for an answer every morning, Rose will be asked every evening. I can see 12 bars. I have more bars than Rose. R has at least 6 bars.
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She is therefore aware that I know she has at least 4. I must give an answer first. Mark will be asked for an answer every morning, I will be asked every evening. I can see 8 bars. Mark has more bars than me. M has at most 12 bars. He is aware that I know he has at most Mark must give an answer first. Morning of Day 1: Mark is asked whether there are 18 or 20 bars? The common knowledge is that Mark sees at most 14 bars. Rose already knows this but also knows that Mark needed to tell her. Now they both know she knows this. The common knowledge is now that Mark has at most 12 bars.
Evening of Day 1: Rose is asked. As Mark can have at most 12 bars, the lowest number of bars that Rose can have is now 6. Mark did not know this. Had Rose seen 6, she would have known there were 18 bars and given the answer. Mark now knows that Rose has 8 bars and has the answer. Morning of Day 2: Mark is asked. Mark gives the answer 20 and both are freed. As some people still have trouble understanding how they can logically figure it out by induction, I'm going to introduce two more people, so that Mark and Rose don't actually know how many bars are on the windows.
Mark will be asking Mary a question, while Rose will be asking Rove. Mark is asked whether there are 18 or 20 bars in total, and he turns to Mary. Rose is now asked, and, knowing that Mark could have answered if there were 18 bars on his window, concludes that there can't be 18 on his window. Therefore, there's one way she could be confident of the total - "Are there 2 bars on our window?
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He says no, and Mary cannot answer the question. Mark knows that Mary will have asked if there are two bars on her window, and as she couldn't answer, he also knows she doesn't have 2 bars. This means there's just one way he could know the answer, and he asks if there are 16 bars on his window. Mary says no, and Mark cannot answer. Now Rose knows there aren't 16 bars on Mark's window, asks if there are four on hers, Rove says no, and Rose can't answer.
This continues - Mark knows Rose doesn't have four, asks Mary if their window has 14, she says no, and he cannot answer. Rose knows Mark doesn't have 14, asks Rove if their window has 6, he says no, and she cannot answer. Mark now knows that Rose doesn't have six, and asks Mary "Do we have 12 bars on our window?
Mark now knows the answer: there must be 20 bars between the two cells, as Rose has at least 8 and Mark has But here's the thing - even if Mark and Rose know the real numbers, rather than needing Mary and Rove to check for them, the logic is still the same. Having the information about the number of bars on their own window doesn't restrict their ability to draw the same conclusions. At each step, there's only one relevant piece of information outside of the known restrictions at least two bars on each, total is either 18 or 20, etc : "is the number of bars on my window the number necessary for me to be able to determine the answer right now?
The introduction of Mary and Rove allows us to see this in action more easily, but is unnecessary for the logic itself.
Mark sees his window with the 12 bars. He can deduce Rose has either 6 or 8 bars in her window. He does not know which. However, he can deduce that Rose will have deduced he has either 10, 12, or 14 bars. Rose sees her window with the 8 bars. She can deduce Mark has either 10 or 12 bars in his window. She does not know which. However, she can deduce that Mark will have deduced she must have 6, 8, or 10 bars.
Most importantly, the knowledge that the other has not answered yields no new information, since they already knew the other wouldn't be able to answer. However, they both know of the strategy which has been laid out by other answerers. They also realise that if they were to follow that strategy, they would get out, so they do. They will never escape. From the info they have and can obtain from each other by knowing that the other hasn't answered, they cannot possibly deduce the number of bars the other one has, hence they cannot possibly deduce the total number either.
Neither Mark nor Rose cannot figure the answer without info from the other one. I hope that much is obvious. Now, the only way for them to send a message to each other is by answering or not answering the question. Mark goes first, he knows Rose has either 8 or 6 bars. He does not know how many, so he does not answer. Rose knows, that Mark didn't answer, and that Mark has either 10 or 12, and that Mark must have guessed Rose has 8 10 or 8 6 and in either case Mark wouldn't have answered.
So Rose didn't receive absolutely no new information from Mark non answering, and cannot answer herself. Mark in turn didn't get anything new from Rose not answering. They cannot escape by using logic, and being genius logicians they do know it. The end.
Now about accepted so called answer.
It has nothing to do with logic because Roses conclusions do not logically follow from the info she has. It is a wrong answer. Downvote all you like. That answer would maybe, sort of, make some sense if the two were allowed to agree on some sort of code beforehand, but I still doubt it. Besides the point anyway. Thank you for your interest in this question.
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Home Questions Tags Users Unanswered. Are there eighteen or twenty bars in my castle? Ask Question. The rules are simple, and solving the riddle is the only hope the two friends have for their salvation: In the castle there are no bars on any window, door or passage, except for the windows in the two logicians' cells, which are the only barred ones this implies that each cell has at least one bar on its window.
The Evil Logician will ask the same question to Mark every morning: " are there eighteen or twenty bars in my castle? If either of them answers correctly, and is able to explain the logical reasoning behind their answer, the Evil Logician will immediately free both of them and never bother them again.
If either of them answers wrong, the Evil Logician will throw away the keys of the cells and hold Mark and Rose prisoners for the rest of their lives. Both Mark and Rose know these rules. Dan Russell Marco Bonelli Marco Bonelli 1, 2 2 gold badges 13 13 silver badges 26 26 bronze badges. If the former, is it guaranteed that one of the options is correct i. Did you come up with it or it is a famous one? The second time I see one of your puzzle it's coincidently also one that has been talked about by the same youtube channel.
Two players, one arbiter Evil Logician. Each player tells only the arbiter how many bars are on the window and the arbiter tells both players only the right sum and a wrong one. We go now step for step through the puzzle and if this really works, it could be demonstrated, right? Nov 15 '16 at The answer: They can escape, and will do so in four days. The explanation: Let's start on the time of release and follow the thought process. Joe Joe 3 3 silver badges 3 3 bronze badges. You cannot draw any conclusions from "if Mark had 18 bars In the hypothetical situation that Rose had 2 bars, Mark having 18 would be perfectly possible.
It may help to think about the simpler cases: what if Mark had 2 bars, Rose had 1, and the question was 3 or 4 total? If you look up the "forehead spot puzzle" or the "blue eyes" puzzle they follow a similar chain of reasoning and may be helpful. But it is not logical. There's nothing logical in making conclusions that do not follow from premises. It's like saying the sky is clear today therefore 20 bars. This may be valid in a context of some secret code agreed upon beforehand, but it has nothing to do with logic.
Assumptions We assume both have at least one bar on their window or the window couldn't be said to be barred, and they're told their windows are the only barred ones. Explanation With those assumptions: Mark knows he has 12 bars. I would have formatted it a bit better though. Thanks for the downvote, whoever. Rose knows both that Mark has 10 or 12, and also that because Mark didn't give an answer that he cannot have more than 17 or fewer than 2.
This second bit of knowledge continues to be refined as time passes with no answer given, until it finally decides between the two possibilities each knows the other can have from the first bit. So they each know what they know, and what the other knows, and what the other knows THEY know. But in this case the numbers just happen to be such that this extra knowledge doesn't actually help them at all. So it's not included in my analysis. Here's where I get hung up: Mark not answering on day one tells Rose nothing she doesn't already know: Mark must have either 10, 12 or 14 bars.
She can not possibly think that Mark has Likewise, Rose not answering on day two does not tell Mark anything he doesn't already know. He knows that Rose has either 8 or 6 blocks, and further, knows that Rose obtained no new information by him not answering on day 1. Mark has no new information on day 3. The cycle repeats. I understand this is probably wrong - but what am I missing? In fact, viewing it another way, the fact that not answering provides "no new information" and in fact doesn't provide any new direct information about the number of bars is exactly why the puzzle is solvable, because It says that the previous answer provided no new information.
The "conversation" the prisoners have amounts to this: Mark: I don't know how many bars there are. Gabriel Burns Gabriel Burns 2 2 silver badges 10 10 bronze badges. Yes, it gets tedious. If anyone can't understand how the non-answers provide information to limit how many bars each knows the other might have, this explanation should explain it entirely. At this point, anyone continuing to argue for unsolvability falls under the umbrella of "There is none so blind as he who will not see.
I imagined a simpler problem: Mark has 1 bar; Rose has 1 bar. The evil logician asks: 2 or 3? The evil logician first visits Mark: Mark knows that he has 1 bar, and that the total number of bars is 2 or 3.
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So he knows that Rose has 1 or 2 bars. Mark says nothing. The evil logician then visits Rose: Rose knows Mark would have said 3 if he had 2 bars. As Rose has 1, and the total is 2 or 3, Rose now knows that Mark has 1 bar; and that total number of bars is 2. She says "2". Fabich Fabich 6, 2 2 gold badges 28 28 silver badges 57 57 bronze badges. Very good answer. Instead, it's about depth of knowledge — think of the initial knowledge like this: Depth 0 : Mark has 12 and he knows it ; Rose has 8 and she knows it. And so on down. To see this, let's consider a different case: Mark has 16, Rose has 2.
In this case, Rose knows that Mark knows that Rose has either 0 or 2 or 2 or 4. On Day 2, Mark is now being tested on his Depth 3 knowledge.
Yes, I know it's getting hard to follow. To get to the final answer, Mark is going to have to determine the truth from Depth 7. In other words Peregrine Rook 4, 2 2 gold badges 20 20 silver badges 39 39 bronze badges. Glen O Glen O 3, 1 1 gold badge 11 11 silver badges 29 29 bronze badges. If you drop to 15 - 3 you are then away from the boundary condition, and you don't get the solution - unless both participants follow an inductive process that doesn't logically follow in reality.
And so it propagates. You realise the person with 3 bars knows the other has 15 or 17 bars, there is then no longer any concept of 18,19 or 20 bars being 'excluded' by a non answer, unless a completely notional concept of them doing this is artificially introduced. And the exclusion process still works - it's just that, rather than being caught when "0" is a possibility at that level, it's caught when "-1" is a possibility. Think about it - if it were 17 and 1, then Mark wouldn't be able to answer, and then Rose would because Rose knows Mark would answer if he had With 15 and 3, it takes a little longer - Mark doesn't answer, then Rose doesn't answer, then Mark doesn't answer, then Rose answers Rose can't answer because she doesn't know whether Mark thinks she has [1 or 3] or [3 or 5].
Mark then can't answer because he doesn't know if Rose thinks he thinks she has [1 or 3] or [3 or 5] or [3 or 5] or [5 or 7]. But then Rose can answer because if Mark had 17, he'd have been able to answer because if he had 17, then [3 or 5] or [5 or 7] would mean the total was at least This is also how I found it, I then could confirm with the previous answers it was the correct one : Let's start by a simpler problem. Here is the illustrated graph: We can note that the values of the nodes in the tree can be computed from the previous values, and Mark and Rose have fixed answers Mark is always telling about 20 for instance.
You get to the right day but by slightly incorrect limiting. Since the result is the same, I went ahead and included it so that my solution is good for any cases. The second one is indeed better, it involves more thinking, and also the Evil Logician says that "the windows in the two logicians' cells are the only barred ones": a barred window must have at least one bar to be considered such, so yeah, the solution is more general than the actual solution to this problem.
So, instead, here are tables of who would say what and when - up to the point at which they free themselves in the current scenario given those two situations: Zero bars is a possible situation: If person has [a, or b] bars they will say "total" and be correct Day Morning Mark Evening Rose 1 [19,20]"20" [0,1]"18"; [19,20]"20" 2 [17,18]"20"; [0,1]"18" [2,3]"18"; [17,18]"20" 3 [15,16]"20"; [2,3]"18" [4,5]"18"; [15,16]"20" 4 [13,14]"20"; [4,5]"18" [6,7]"18"; [13,14]"20" 5 [11,12]"20" Note that For the "zero is possible" case: On the first evening if Rose has 0 bars and is not free, she knows that Mark does not have 20 bars and hence may deduce there are 18 in total.
Jonathan Allan Jonathan Allan So this answer is wrong actually. That makes things more narrowly confined throughout. In this specific case it ends up not mattering, but for different numbers it would have been the difference of a day. Breakdown of Events: Morning of Day 1: Mark is asked whether there are 18 or 20 bars? SeaMonkey SeaMonkey 5 5 bronze badges. An empty mind is the ideal state to have when you are faced with a pressure putt. However, the ability to clear negative thoughts from entering into our consciousness is entirely another matter.
Most of us are not very good at it. Visualising a putting path instead of a line makes good sense as a path is thought of as being WIDER than a line. In your mind the task of rolling your ball along a broad path will simplify your putt. Straight Putts are not the easiest of putts as golfers have a tendency to pull or push their putts.
In fact the ability to hit your ball end over end in a straight line is a skill that you must learn. Articles 90 to Putting Excuses are a convenient way of explaining away a poor putting round. Self-deception may ease your disappointment, but the truth is that any failure is invariably down to your lack of skill. Excellent Putting is not about holing every makeable putt. That would be unrealistic. It is about keeping the percentages in your favour. You should strive for excellence, but never expect perfection. Downhill Putts are the putts most deserving of your attention.
Of all the putts they are the most difficult to accurately judge both the speed and the break. They lead to many three-putts. In fact, once you are lined up on the green, sight can interfere with your focus. Outside distractions can catch your eye. Better Putting is what you should aspire to.
It is essential for you to build up a smooth repeating stroke. Patchy Conditions on the greens are part of the challenge of golf. The greens have to be maintained and there will be times when the putting surface is less than ideal for the ball to roll smoothly on line. Ball Speed plays an important part in shaping the roll of your ball on the putting surface. You can't control the slope or speed of the green, but you can control how hard or soft you hit your ball. The pendulum stroke is considered by many to be the most reliable way to execute a putt.
However, strictly speaking such a stroke is not a true pendulum as there is no fixed pivot to guide your swing. The Putter Butt refers to the top of the shaft. During your stroke it should retain the same relationship to your arms, hands and wrists as established in your setup. But where should it point? Your Aiming Preference will dictate if you are more comfortable picking out a specific spot on your target line, or prefer to trust your natural instincts to get your ball rolling on line.
Articles 80 to Aiming Method Z is a useful technique when you have a mid to long range breaking putt. It is a popular aiming method of aiming at the Apex of the Break. Aiming Method Y is a technique for breaking putts where you pick an aimpoint to the side of the hole. The idea is to imagine a number of golf balls even with the hole, and then aim for one of them. For the average golfer this could be for putts within the six foot. Breaking Putts, either left to right or right to left, are more common than straight uphill or downhill putts. The question is whether you should treat each of these putts slightly differently?
Alignment Aids have taken on a new life with the popularity of mallet putters. With the larger putter head, designers have more space behind the putterface to experiment with different shapes. Confidence, cynically described as the state of mind before you understand the problem, plays an essential part in your ability to putt well under pressure.
However, it is better described as a feeling of assurance. Modern Putter Designs are more the creation of marketing efforts to outsell competitors than a concerted attempt to engineer a putter that will assist in making putting easier for the average golfer. Your shoulders play an important role in guiding your putter. Their ball and socket construction allows you to move them in different ways. The question is whether you should rotate them or rotate them in your putting stroke? Eye deception is a frequent problem in putting. Where you think you are aimed is not where you are actually aimed.
A perfect stroke with a flawed aim translates pretty much every time into a missed putt. Putting in sync entails matching your stroke to your posture, and then finding a putter that complements your stroke. The best results comes when there is a perfect match. Articles 70 to Should your putter choice determine your putting style, or is it the other way around?
This is a putting dilemma that every golfer faces - unfortunately getting the answer wrong most of the time. What Body Height should you adopt when over the ball in putting? Should you crouch, stand upright, or find a position somewhere in between? Putting is a mind game as well as the physical act of propelling your golf ball across the green.
However, how you perceive reality will be determined more by competence than by any putting magic. The Belly Putter has been the salvation for some golfers whose putting has gone south. However, there is still the thought attached to its use that it is a band-aid solution rather than the real deal. What part does your Right Hand play in the putting stroke? If you are right-handed, it is normally your dominant or master hand. It is the hand that does most of the work in everyday tasks. However, in putting should it have a controlling role?
A split grip is said to increase the control over your putter and allow you to putt with greater accuracy. It was used as early as the nineteenth century so the idea that it is something new is wrong. A Cut Stroke in putting is where the path of your putter travels through the ball from outside the target line to inside the target line.
The question is: Why does this matter as long as your putterface is perpendicular to your target line at impact? Putting Short is not only frustrating, it also adds to your score and deprives you of useful information. If you miss long, at least you learn which way your ball will break on the return putt. Depth Perception. Putting is not just about stroke mechanics and technique. Your vision and depth perception play a vital role in seeing slopes on the green, aligning your putterface accurately, and judging distance. Arched wrists position the putter shaft so that it is better aligned with your forearms.
This helps to remove unwanted wrist action in the form of hinging up and down from your putting stroke. Articles 60 to Putting Smart is all about putting in such a way that everything in your setup and stroke helps rather than hinders your ability to make a good putt. Why make putting more difficult than it is? Putting on Plane is the ability to keep your putter on plane up to the moment of impact. At setup you aim your putterface to your target.
If your putter subsequently moves off plane in your backstroke, it will be difficult for you to re-square your putterface on your forward stroke. It derives from the belief that the best putter path moves inside the target line on the backstroke, is square at impact and then returns inside again on the through stroke. The Angle of Contact is the angle at which your putterface contacts the ball at impact in your putting stroke. If the angle on approach is too steep, or too shallow, you will have trouble with both your direction and distance control.
Your left elbow forms part of the putting triangle that is recommended for the putting stroke. However, within this framework you have a degree of choice as how to manage its position. Putting Comfort at the expense of proper form can trick you into setting up to the ball in such a way that you make it more difficult for yourself to sink your putts consistently. There is a saying that no golfer has ever looked up from a shot and seen a good one. The same saying can be applied in spades to putting when you look up too soon in your follow through. Eyes Down is the usual address setup when you are putting.
The question is where exactly should you look? The answer to this question depends on who you listen to. The Wrist Angles that you establish at address are critical to the impact position of your putter during your putting stroke. Change them mid-stroke and you could be in trouble. Toeing the putt is a common suggestion for handling fast downhill putts, or putts that break sharply right-to-left.
The question is whether this is a good idea or just a compromised technique? Articles 50 to A Bladed Wedge is a short-game technique that pros often use when their ball has nestled against the collar of the green. It is a smart choice as you have more chance of contacting the ball cleanly. Reading the Break starts with gauging the speed at which your ball is likely to roll enroute to the hole. Except for straight up and down putts on the Fall Line, all putts break one way or the other. Golf is an outdoor game and windy conditions can play havoc with your putting if you let the wind get the better of you.
However, it does not mean you have to putt poorly. Fast greens are not what every golfer needs, or should ask for, in order to putt better. In fact, the exact opposite is the case as fast greens soon lead to fearful putting. Solid Contact with the ball is a critical component in distance control on the green.
Without it you are going to three-putt too many times for your liking. The Putting Decision you must make on every putt is crucial to the outcome. Get it wrong and you could be looking at a three-putt green. Make the right choice and the worst you can do is two-putt. The Putting Game stands in direct contrast to the rest of your shots from tee to green.
Nothing that you learn on the driving range is readily transferable to the putting green. Golf is a game that lasts a lifetime. However, your Putting Technique will haunt you in your advancing years unless it is based on sound fundamentals. You should therefore act now before it is too late. Putter Acceleration on the downstroke is essential for accurate distance control. However, this does not mean consciously speeding up your stroke as the putter head approaches the ball. Putting Tempo is the total time your putting stroke takes from start to finish. Different golfers have different tempos, but the tempo of most golfers is generally too quick.
Articles 40 to The Reverse-Overlap grip has been the recommended putting grip for a long time. Today it is under threat. This is because the best way to hold the putter has become anyone's guess. Setup at Address. A consistent Setup at Address is an important fundamental in putting.
It does your putting no good to keep changing the way you stand to the ball. You need to do the same thing every time you putt. Judging distance is a task that is accomplished largely by sight. However, most of us are not very good at it. While vision is important, it is not the only source of information for judging distance. Emotional Putting is an unhelpful habit whereby we label our putts.
As soon as you start to label your putts as anything more than just another putt, you start to put pressure on yourself. The choice among flatsticks is a moving target as manufacturers jostle for market share. You need to think hard before you commit to a change. Is it really your putter that is hurting your putting? Speed Putting is a technique to free up your putting and turn it more into a reactionary motion. Unfortunately most golfers dwell over their putts too long allowing doubt and tension to creep in.
Only fools believe that you can buy a better game of golf. You are truly missing the point if you think that a new putter is the only thing that stands between you and greater success on the green. Continuous putting is an unofficial rule to speed up play. It doesn't mean rushing your stroke.
However, missing a short putt through carelessness will undermine your confidence well after the round is over. Better long putting leads to lower scores. This means doing what you can to avoid three-putting in the round. Using your putting stroke with a chipping setup helps when faced with a long putt.
Putting Stats are a curiosity, but often of little practical value to improving your putting. If you don't know if you are putting well or badly without them, you are just not paying attention. Articles 30 to A positive Mental Image is a clear picture in your mind's eye of a successful outcome. For example, on the green you can see your ball tracking to the hole, and then hear it falling in. True roll of the golf ball off the putterface is one of several marketing promises made to golfers hoping to buy a better putting game.
In reality, putter design using the latest anti-skid groove technology plays only a small part in achieving the promise of true or pure roll. Making the Putt compared to Holing the Putt may just seem like a play on words, but it is not. It is a completely different mind-set.
It is an attitude that will keep your emotions in balance. Many golfers don't realise how much their body moves when putting. This swaying movement unsettles their balance and causes their centre of gravity to shift about during their putting stroke. The objective on every putt is to strike the ball when your putterface is perpendicular to your aimline.
This will be difficult to do if you allow your left shoulder to rotate backwards during the downstroke and a little beyond. Your anatomy attests to the fact that you were born to move. However, to putt well you need to be still. Unnecessary movement during your putting stroke is a killer of consistency. Swearing and cussing intrudes on our everyday life. We hear bad language everywhere and, heaven forbid, even on the golf course. But can swearing and cussing help to improve your putting? Many golfers use the line on ball technique to aim when putting.
They assume that because they are aiming from behind, their aim is accurate. Unfortunately this is often an illusion of accuracy. The Clock Drill is an excellent way to practise short putts while learning how to read the break of a putt. As you move around the clock, you will see that your ball breaks in different directions. To putt well you must be able to putt straight. If you can't hit your ball in a straight line over a short distance on a true and level surface, you will have difficulty in becoming a good putter.
Articles 20 to Sticking your tongue out when putting makes more sense than you would think. If you watch closely the faces of professionals when they are concentrating, you will see a number of them stick their tongue out to the side of their mouth. Your right elbow may be more important in putting than you think.
Your right elbow can anchor your backstroke in much the same way as affixing the butt of your putter against your body. Breathing is not something that you have to think about in everyday life. It is an automatic body function. However, breathing can be controlled consciously. Is the width of stance in putting immaterial? If you were asked for your opinion on how to set up your feet what would be your recommendation?
A forward press is a technique that some golfers use to initiate their putting stroke.