puzzles

Geocaching Like Batman

As the days get longer and the weather gets better, woodland trails and city alleyways alike begin to more-frequently see a particular brand of explorer. Clutching GPS devices (or, increasingly, mid- to high-end mobile phones), these satellite-guided adventurers shy away from normal people, whom they call “muggles”. Outwardly, this is out of concern for the continuity of their tiny treasure, but as often as not, it’s because geocachers – and especially urban geocachers, who often don’t even have the excuse of “getting some fresh air” to justify their hobby – are likely to be seen as a little odd., “You do what for a hobby? Find lost lunchboxes?”

I’ve written plenty about geocaching already, but the only important thing to know for this particular anecdote is how geocaches are rated to indicate how hard they are. There are two scales, each scored from one to five “stars”. The first scale is difficulty, which is about how challenging the geocache is to find – a 1-star rating means that it’s in plain sight, not camouflaged, etc., while higher ratings might mean that it’s well-concealed, tiny, disguised as something else, or requires that you solve a puzzle in order to determine where it is. The second scale is terrain, which is about how challenging the geocache is to get to. A 1-star rating is typically accessible by wheelchair – you certainly don’t need to leave paved roads and footpaths to get it; higher ratings might mean steep gradients, tree climbing, long hikes, and so on. The highest terrain ratings often mean that specialised skills or equipment are required (for example, rock climbing gear or a scuba tank).

As you can imagine, caches with a 5-star “terrain” rating are rarer, and are especially uncommon in built-up areas. Half-way up cliffs… deep inside caves… miles out to sea: these are the places you’d expect to see geocaches with the highest level of “terrain” score. So imagine my surprise when I discover GC13WZQ (“Swing Lower”), a geocache with a 1-star “difficulty” rating but a 5-star “terrain” rating, just a few minutes walk from Oxford City Centre. In the seven years this cache has been in place, it had seen fewer than 110 successful visitors: contrast to its neighbour, GCK57Z (“Swing Low”) – a virtual cache less than 10 metres away – which has seen about six times as many visits in only 3 years longer. This, I thought, was a cache I had to see.

The sides of the bridge, boarded up and with barbed wire. Photo by OxfordLad on Geocaching.com. — OxfordLad (who took this photo), and other geocachers claimed that, since early 2014, the cache was made entirely inaccessible by the boarding-up of the sides of the bridge.

Folks recently attempting to find the geocache had reported (OxfordLad, izybuzyfingers, twitcher50) that it had been made inaccessible by the recent addition of boards and barbed wire to the edges of the bridge. Counter-arguments were raised (sandvika, Mad H@ter) to show that this didn’t make the cache inaccessible; it merely made it accessible only by boat, which had already been suggested in the “attributes” for the cache.

Geocache with "boat" attribute — Only an idiot would attempt a ‘requires boat’ geocache without a boat. Right?

I’m not a believer in the idea that any particular geocache can only be found one particular way. Also: I don’t have a boat. So I decided to make an expedition to “Swing Lower” my own damn way. Approaching the bridge under which the cache is located, I immediately saw the boards and barbed wire that had been reported by those that had attempted it earlier in the year. But as I would soon discover, anybody who was put off by a little bit of plywood and the risk of damp feet really wasn’t built of the right stuff to be able to do what was required next. Put simply: boards and barbed wire are the least of your problems when you’re hunting for GC13WZQ.

The bigger challenge was getting to the cache once underneath the bridge. I discovered (perhaps with a little inspiration from “Jackhuber”) that it was possible to brace myself against a pair of the beams that run the length of the bridge and – facing down – shuffle sideways to get to the centre of the bridge. I felt acutely aware of the fact that until I got over the central channel, the depth of the water might not be enough to break my fall (especially if I slipped and fell head-first), but was reassured by the fact that I’d brought fellow ‘cacher and coworker kateevery and she was ready, perhaps not to swim out and get me but at least to call 999, should the need arise.

Dan holding the geocache he's found above his head, triumphantly. — This is how Freddie Mercury holds a geocache.

So there you go. To all of you wusses for whom “there are boards and barbed wire in the way” was an excuse: you hadn’t even begun to face the challenge of “Swing Lower”. I’ve written up a Batman-themed description of the expedition as part of my log report.

A screenshot of the clue for GC54F78, one of the caches in my new series. — Can you make out the coordinates in this image? No? Maybe it’d help if you looked at geo.danq.me.

This conveniently coincides with the week that I launched my new collection of puzzle geocaches, the Oxford Steganography Series – four geocaches (GC54F78, GC54F7B, GC54F7J, GC54F7N) whose coordinates are concealed within images or text, each of which contains a transparency film that can be used (I made a video showing how) to determine the coordinates of a fifth, bonus cache. I’m reasonably pleased with the series, and I’ve been enjoying reading the reports of the ‘cachers who’ve been out hunting for them, so far.

Suppose you have a time machine that can only jump to leap days. What’s the chance that a random jump will put you on a Monday? [Maths]

This link was originally posted to /r/puzzles. See more things from Dan's Reddit account.

The original link was: http://www.scatmania.org/2012/09/24/leap-machine/

Here’s a puzzle for you –

Like the TARDIS, your time machine has a fault. The fault isn’t a failure of its chameleon circuit, but a quirk in its ability to jump to particular dates. Picture courtesy aussiegall (Flickr), licensed Creative Commons.

You own a time machine with an unusual property: it can only travel to 29^th February. It can jump to any 29th February, anywhere at all, in any year (even back before we invented the Gregorian Calendar, and far into the future after we’ve stopped using it), but it can only finish its journey on a 29^th of February, in a Gregorian leap year (for this reason, it can only jump to years which are leap years).

One day, you decide to take it for a spin. So you get into your time machine and press the “random” button. Moments later, you have arrived: it is now 29^th February in a random year!

Without knowing what year it is: what is the probability that it is a Monday? (hint: the answer is not ¹/₇ – half of your challenge is to work out why!).

Dan Q

The Leap Machine (Puzzle)

Here’s a puzzle for you –

Like the TARDIS, your time machine has a fault. The fault isn’t a failure of its chameleon circuit, but a quirk in its ability to jump to particular dates. Picture courtesy aussiegall (Flickr), licensed Creative Commons.

You own a time machine with an unusual property: it can only travel to 29^th February. It can jump to any 29th February, anywhere at all, in any year (even back before we invented the Gregorian Calendar, and far into the future after we’ve stopped using it), but it can only finish its journey on a 29^th of February, in a Gregorian leap year (for this reason, it can only jump to years which are leap years).

One day, you decide to take it for a spin. So you get into your time machine and press the “random” button. Moments later, you have arrived: it is now 29^th February in a random year!

Without knowing what year it is: what is the probability that it is a Monday? (hint: the answer is not ¹/₇ – half of your challenge is to work out why!).

Webcomics With Puzzles

Like puzzles? Like webcomics? Then here are two things you ought to see:

Crimson Herring

The first is the short-lived webcomic Crimson Herring. Personally, I’m hoping that it’ll come back to life, because it really had lots of potential. In each episode, a “crime drama” plays out, and you – the reader – are left with just enough clues to solve the case. Sometimes you have to really pay attention to the pictures, other times to the words, and it’s really got a good idea going for it.

Even if it turns out to be completely dead, now, you can go back and read the archives: start here! And if you like it, leave a comment and let the author know; see if we can get it brought back again.

A recent Abstruse Goose

A recent Abstruse Goose, called “A Simple Puzzle 4”, had me thinking for a few days, and then the answer suddenly came to me.

Frame 29 from Abstruse Goose - A Simple Puzzle 4. — Frame 29 from Abstruse Goose – A Simple Puzzle 4.

The idea behind the comic is really quite clever; but once you’ve worked out the key, putting the panels into the right order isn’t difficult at all. Give it a go!

A Primer Puzzle

There’s a film that I’m a huge fan of, called Primer. Since I first discovered it I’ve insisted on showing it at least twice at Troma Night (the second time just for the benefit of everybody who didn’t “get it” – i.e. everybody – the first time). If you haven’t already seen it, this post might be a little spoilery, so instead of reading it, you should warm up your time machine, go and watch the film, turn off the time machine, get into the time machine, come out again right now, and then read its Wikipedia page until you understand it. Then come back.

Still with me? Right.

Why Primer is awesome, and why you should care

In Primer, the protagonists accidentally stumble across the secret of time travel and use it to cheat the stock market. The film isn’t actually about time travel or science-fiction: it’s actually about the breakdown in the relationship between the protagonists, but it’s got some pretty awesome science-fiction in it, too, and that’s what I’d like to talk about. The mechanism of time travel in Primer, for example, is quite fascinating: the traveler turns on the machine using a timer switch (turning it on in person risks the possibility of meeting a future version of themselves coming out of the machine). They then wait for a set amount of time, then they turn off the machine, get into it, wait for the same amount of time again, and emerge from the time machine at the moment that it was turned on.

This is a lot weaker than many of the time travel devices featured in popular science fiction literature, films, and television. It’s not possible to travel forwards in time (except in the usual way with which we’re familiar). Travelling backwards in time takes as long as it took the machine to travel forwards through the same period, making long journeys impossible. The machine has to be strategically turned on at the point at which you want to travel back to, reducing spontaneity, and it can’t be used again in the meantime without resetting it. Oh, and the machine is dangerous and causes long-term damage to humans travelling in it, but that’s rather ancillary.

There’s a certain believability to the time travel mechanic in Primer that gives it a real charm. As far as it is explored in the film, it permits a deterministic universe (so long as one is willing to be reasonably unconventional with one’s interpretation of the linearity of time, as shown in the diagram above), provides severe limits to early time travel (which are great for post-film debate), and doesn’t resort to anything so tacky as, for example, Marty McFly gradually “fading out” after he inadvertently prevented his parents from getting together in Back to the Future.

Experiments in the Primer universe

I’ve recently been thinking about some of the experiments that I would be performing it I had been the inventor of the Primer time machine.

First and foremost, I’d build a second, smaller time machine of the same design. We know this to be possible because the first machine built by the protagonists is smaller than the ones they later construct. I want to be able to put one time machine inside another. Yes, yes, I know that this is what the protagonists do in the movie, but mine has a difference: mine is capable of being operated (power supply only needs to be a few car batteries, as we discover in the film) within the larger time machine. That’s right, I’m building a time machine inside my time machine.

Experiment One attempts to explore the relativism of time. Start the larger time machine and warm it up. Stop the larger time machine. Start the smaller time machine. Get into the larger time machine, carrying the smaller time machine, and travel back. Once back, turn off the larger time machine. Experiment with sending things forwards in time using the second time machine (which has traveled backwards in time but while running, from our frame of reference). If objects inserted into it come out in the future, before it is picked up, this implies that there might be a fixed frame of reference to chronology. It also indicates that it is possible to build a machine for the purpose of traveling forwards in time, too, although only – for now – at the usual rate.
Experiment Two attempts to accelerate the rate at which a traveler can move forwards or backwards through time. Based on the explanation given in the movie, the contents of the time machine oscillate backwards and forwards through the period of time between their being turned on and being turned off, for a number of repetitions, before settling. If we are able to synchronise the oscillations of two time machines, one inside the other (by turning them on and off simultaneously, using timers attached to each and their own distinct, internal, power supplies), might we be able to set up a scenario that, in X minutes, switches off, and we can get inside the inner machine and travel back to the switch-on time in X/2 minutes? If so, what happens if we send such a two-machine construction back in time as in Experiment One – do we then have a “time accelerator”?
Experiment Three takes advantage of the fact that for an object within the field, an extended period of time has passed (during the oscillations), while from the reference point of an external observer, a far shorter period of time has passed. Experiment with the use of an oscillating time period field to accelerate slow processes. Obvious ones to start with are the production of biologically-produced chemicals, as is done in the film (imagine being able to brew a 10-year-old whiskey in a day!), but there are more options. Processing time on complex computer tasks could be dramatically reduced, for example. Build a large enough time machine and put a particle accelerator in it, and you can bring masses up to relativistic speeds in milliseconds.
Experiment Four is on the implications on spacetime of sending mass back in time. As we know, flinging mass in a direction of space produces an equal and opposite acceleration in the opposite direction, as demonstrated by… well, everything, but let’s say “a rocket” and be done with it. Does flinging mass backwards through time produce an acceleration forwards through time? This could be tested by sending back a mass and a highly-accurate timepiece, removing the mass in the past, and letting the timepiece travel back to the future. The timepiece is checked when the experiment starts, when the mass is removed, and when the experiment ends. If the time taken for the second half of the experiment, from the perspective of the timepiece, is longer than the time taken for the first half, then this implies that Newtonian motion, or something equivalent, can be approximated to apply over time as well as space. If so, then one could perhaps build an inertia-generating drive for a vehicle by repeatedly taking a mass out of one end of a time machine, transporting it to the other, and sending it back in time to when you first picked it up.

The scientific possibilities for such a (theoretical) device are limitless.

But yeah, I’d probably just cheat the stock market, too. At least to begin with.

The Three Demons Puzzle

Spent my entire lunch break solving this brainteaser that some sadist e-mailed to me, so I thought I’d share it with you. I’ll post a solution soon.

The Three Demons Puzzle

You have been granted an audience with the three demons of time and space, who know everything about the past, present, and future, and can even read minds: Amos, Baeti, and Corpi. You are allowed to ask them only three questions, but you can direct these three questions at the demons in any configuration: so you could, for example, ask all three questions of one demon, if you wished. Obviously this gives you a great deal of power, and you could use it to learn any secret you desired, but, as always, there is a catch:

The demons will only answer questions that can meaningfully be answered with “yes” or “no”.

Still; that’s not so bad. But it gets worse:

One of the demons always tells the truth.

One of the demons always lies.

One of the demons randomly answers “yes” or “no”, regardless of the question asked.

You do not know which demon is which.

Starting to get a bit more problematic? There’s more:

The demons will only answer in their native tongue, saying “da” and “ja” rather than “yes” and “no”. You do not know which demon syllable (“da” and “ja”) means which answer.

The aim of the puzzle is to determine which demon tells the truth, which one lies, and which one is random.

Special Rules And Tips

Some thoughts to help you get started and to ensure you don’t accidentality cheat:

You may only ask “yes or no” type questions. For example, you could ask Amos “Did Baeti say ‘yes’ to the last question I asked of him?” but you could not ask “What answer did Baeti give to the last question I asked of him?” Despite the fact that the latter would be expected to produce the same result as the former, the context is different: all questions must be phrased as “yes or no” type questions.

There is no point in repeatedly asking a demon the same question in order to try to determine whether or not he is the one that answers randomly.

If you can’t find a solution, try first removing the “you may only ask three questions” restriction. My first solution required that four questions be asked, for example, and I later refined the first two questions into a better single question, once I knew what I needed to determine before asking the next one.

Your Questions

In answer to some of the questions I’ve been asked:

Each question is asked to exactly one demon: you can’t ask a question to multiple demons at the same time.

The answers given by the random demon are random insofar as it is not possible for any human to determine what an answer would be in advance. However, as the demons themselves are able to see the future, each demon would theoretically know what the next answer that the random demon was going to give. However, I can’t think of a way that could possibly be useful.

Good luck! I’m not sure whether or not this is harder than the blue eyes/green eyes puzzle I posted to my blog last year. You decide.

I’ve published the solution in a separate post.