{"id":36,"date":"2011-08-02T02:16:34","date_gmt":"2011-08-01T23:16:34","guid":{"rendered":"http:\/\/www.nicau.ro\/?p=36"},"modified":"2013-04-30T19:41:51","modified_gmt":"2013-04-30T16:41:51","slug":"the-monk-and-the-mountain","status":"publish","type":"post","link":"https:\/\/www.nicau.ro\/?p=36","title":{"rendered":"The monk and the mountain"},"content":{"rendered":"

\"\"<\/a><\/p>\n

The other day, while talking to a friend he asked if I was up for a puzzle. As I always enjoy a good challenge I answered: what puzzle? Then he started telling and telling, where in the end there was this simple question. From the very start I had an answer but it was restraining the generality, therefore unacceptable. After a bit of analyzing, I have found something much better along with some pitfalls for the careless solver. In this article we will discuss the pitfalls and of course, the *solution* .<\/p>\n

First, let’s have a look at the puzzle:<\/p>\n\n\n\n
A monk starts one morning a journey from the base of the mountain to a monastery at the top. He leaves at 8am and gets there at 5pm. He spends night at the monastery and the next morning he climbs down retracing the route. He leaves at 8am from the top and arrives at the base at 5pm. Is there such moment (of the day) when he’ll be at same location in both days?<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n

 <\/p>\n

SPOILER WARNING><\/strong> If you wish solving this puzzle independently stop reading here.<\/p>\n

 
\nAt a glance, the naive answer is – sure, it is possible as long as he’s going with constant speed, both days. The moment will be around noon, more precisely 12.30pm ( 8 + ( 17 – 8 ) \/ 2) at half distance. This answer induces a new parameter, the speed being constant which affects the question generality, therefore we might as well ignore it.
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\"Google<\/center>
\nSo what is the right answer? Let us read the problem again. Text seems to be straightforward: 8am at the base, goes walking with variable speed. Maybe he starts with great speed covering 6\/10 of distance in just 1\/3 of time, then for the last 4\/10 incredibly slow, enjoying the scenery, consuming the remaining 2\/3 of time. He’s reaching summit at 17. We can think in terms of functions. On the X<\/b> we’re representing time<\/b> from 0 to 9, where we add to 8am to obtain the absolute hour. On the Y<\/b> axis we’re representing the distance from start<\/b> from 0 to 10. The problem does not state the exact distance from base to summit. That’s not important.
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\"Google<\/center>
\nOn the next day he’s taking the same route but starting at 8am from the summit. Let us assume he’s going relatively slow, watching birds with binoculars or something. After 2\/3 of time he’s covered just 1\/10 of distance. When he realizes how late and how much more distance, knowing how mountains are not safe after nightfall, starts racing down. Back in the day the monk used to be athletic, doing mountain marathons so he’s reaching base at precisely 17.00.
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\nAll good so far. We have read most of the text, taking two extreme examples and representing them on such nice charts :). What about the question? It is suggesting somehow, no matter what variables, there should exist a same hour each day when monk’s location will be the same<\/b>. How could it be? In the above examples, the monk is at very different locations at say, 10am – which is 1 hour offset from the start. And “start” means the first day close to the base, the next day close to the mountain top. Doesn’t this look like a valid counter-example, sufficient to answer the puzzle with “NO, it is impossible”? This is the first trap. Actually, no. The puzzle question asks whether there is such moment in time (if it exists<\/b>) and not if every hour meets the criteria. Finding a certain hour where the criteria doesn’t fulfill is wrong. Instead of proving universal quantification we need to prove existential quantification.
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\nIt might be a good idea representing both ascending and descending functions on the same graph. For ex. let us name f<\/b> the ascending function and g<\/b> the descending function. Each function take x<\/b> as elapsed time and produce y<\/b> as distance from the start. Can you think of a condition for both functions to test the puzzle question? Yes, you’re right. We need to put the two in a system of equations:
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\ny = f(x)
\ny = g(x)
\n<\/code><\/center>
\nSame x time of the day produces same y monk location. Let us have the ascending and descending functions drawn on a same graphic.
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\n\"Google
\n<\/center>
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\nWe see no intersections other than at 8am and at 17pm. But then we know it’s impossible for the monk to be in the same location. At 8am in the first day he’s at the base of the mountain, the next day he’s at the monastery. Wasn’t this graph supposed to show us the solution? Where is the error? …And this is how we find the 2nd trap :). Anybody not paying attention could fall for it. There is nothing wrong with having intersections at start and at finish. On Y<\/b> axis we’re representing distance from the start<\/b>. Situation where intersections have some other meaning — the monk does start at 8am (0 offset on X<\/b> axis ) and he does reach destination 9 hours later.
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\nThe question is what we choose to represent on Y<\/b> axis. To be meaningful for the puzzle-question, we should have represented absolute location<\/b>. “..when he\u2019ll be at same location in both days?” – same location can only be referred to as in absolute terms, or needs to be computed. We’re taking the easier option. The ascending function stays the same, he starts from 0 and reaches mountain top at location 10. We need to modify the descending function which given changes will start at location 10 and as time passes, reach location 0.
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\n\"Google
\n<\/center>
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\nCONCLUSION><\/b> Just by watching the above graphic we understand some takeaways. The monk retraces his steps. The trail is uninterrupted. Continually, his current location is a place he has visited both days, only at different moments of the day. For each location, we compute a positive difference between the moments of the day he’s passed that location. Value range for this amount looks like a reversed bell. It’s maximum for locations close to base or summit — almost 9 hours. But as he’s walking towards midpoint, the amount is decreasing to a point where is zero. That’s the location\/moment we’ve been searching for.
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The answer is obvious now — YES.
 <\/p>\n","protected":false},"excerpt":{"rendered":"

The other day, while talking to a friend he asked if I was up for a puzzle. As I always enjoy a good challenge I answered: what puzzle? Then he started telling and telling, where in the end there was this simple question. From the very start I had an answer but it was restraining […]<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":[],"categories":[5],"tags":[],"_links":{"self":[{"href":"https:\/\/www.nicau.ro\/index.php?rest_route=\/wp\/v2\/posts\/36"}],"collection":[{"href":"https:\/\/www.nicau.ro\/index.php?rest_route=\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/www.nicau.ro\/index.php?rest_route=\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/www.nicau.ro\/index.php?rest_route=\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/www.nicau.ro\/index.php?rest_route=%2Fwp%2Fv2%2Fcomments&post=36"}],"version-history":[{"count":158,"href":"https:\/\/www.nicau.ro\/index.php?rest_route=\/wp\/v2\/posts\/36\/revisions"}],"predecessor-version":[{"id":339,"href":"https:\/\/www.nicau.ro\/index.php?rest_route=\/wp\/v2\/posts\/36\/revisions\/339"}],"wp:attachment":[{"href":"https:\/\/www.nicau.ro\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=36"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.nicau.ro\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=36"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.nicau.ro\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=36"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}