Inspired by Infinite Precision

No, I'm not picky, I just want things done exactly right.

Complete 5 of these tasks (tell me how you would do them) and I will give you exactly 25 bragging points. I will not allow any trivial answers ( 0, {}, a straight line is not a square wave, the absence of light is not white, etc ).

  • Excite me a wave which is exactly square.
  • Draw me an exact circle.
  • Brighten my night with a light that is exactly white.
  • Show me a container that is exactly full.
  • Bind me a book which is exactly endless. (Reader will always read left to right, top to bottom, one page after another)
  • Make me an apparatus which rotates exactly once per year.
  • Find me two objects of exactly the same color.
  • 4
    $\begingroup$ I want to concentrate but exactly written in bold diverts my mind. $\endgroup$ – wrangler Jun 11 '16 at 6:43
  • $\begingroup$ @humn no language tricks here, this is all science, so I added the science tag. Looking at the original post might give you some ideas. $\endgroup$ – Tony Ruth Jun 11 '16 at 20:41
  • $\begingroup$ With respect to binding a book that is exactly endless, I would respectfully direct you to John Barth's solution "Frame-Tale" described here. It is pretty cool that he foresaw you asking this question back in 1968. $\endgroup$ – user29544 Aug 29 '16 at 14:21
  • $\begingroup$ @TonyRuth Inazuma, Ankoganit, Lashit, and I all have a list of exactly five items that can be done, and they're not the same list. So does that mean there's an oversight in your puzzle or at least three of our answers? $\endgroup$ – user24580 Aug 31 '16 at 15:43
  • 1
    $\begingroup$ @Tony So can you give us some pointers on how to improve our answers to try to match what you were thinking? Which parts of our answers are wrong? It's hard to answer a puzzle with no clue of how we're doing. $\endgroup$ – user24580 Sep 1 '16 at 2:14

Three ideas:

Make me an apparatus which rotates exactly once per year.

Earth rotating around the sun? Is this an apparatus? But this is the definition of a year.

Show me a container that is exactly full.

<div id="container" style="padding:0;"> <div id="stuff" style="margin:0;width=100%;height=100%;"> Stuff </div> </div>

The container is exactly full (tell me if I am missing something).

Find me two objects of exactly the same color.

In JavaScript:
var a = "object1"; var b = "object2"; a.color = "#f00"; b.color = "#f00";

a and b have exactly the same color.

Another possibility:

You can just take any object and show it twice as it's not mentioned that the objects are different. :P OK, that's probably cheating as the OP asks for two objects.

But this could work:

You could take two atoms because single atoms have no color.

  • $\begingroup$ Yes, this is definitely the right idea for the first two. The rotation uses the definition of a year, and the only containers that can be full which I can think of are digital. As far as the color goes, I did not specify that they had to be physical objects, so I guess this counts. But, I think you can find two physical objects of the exact same color as well. $\endgroup$ – Tony Ruth Jun 11 '16 at 23:31
  • 2
    $\begingroup$ @Tony Ruth How do you "make an Earth rotating the sun"? Show me an apparatus, yes, but make - no. $\endgroup$ – Inazuma Jun 12 '16 at 0:09
  • $\begingroup$ @Inazuma, any object you build on Earth will revolve exactly once around the sun per year. Of course the earth rotates on its axis so any object on the surface of the Earth would pickup additional rotations. But, I think you can come up with some mad scientist contraptions that would rotate when the earth goes around the sun, but not when the Earth rotates on its axis. $\endgroup$ – Tony Ruth Jun 12 '16 at 2:43
  • $\begingroup$ @TonyRuth I suppose that makes sense, but then you have six completed tasks. Also your original post says rotates, not revolves. $\endgroup$ – Inazuma Jun 12 '16 at 2:54

Make me an apparatus which rotates exactly once per year.

A Foucault's pendulum at the north pole will precess one revolution per day. One at the equator will not precess at all. Therefore, there is some latitude north of the equator where the precession rate will be equivalent to one revolution per year.

  • 5
    $\begingroup$ s/North/South/g if you live in the southern hemisphere and are offended by this answer.. $\endgroup$ – Chris Cudmore Jun 13 '16 at 20:59

partial answer: rotates exacty once per year:

the month field of a real-time clock (computer hardware) rotates exactly once per year, when it changes from 1 to 2 this is exactly the same as a rotation left by 1 bit. none of the other annual changes are exact rotations.

two of the same exactly the same colour

two similar sodium low pressure sodium vapour lamps will have indistinguishable spectra, two helium-neon, or carbon-dioxide lasers same.

two pure samples of a coloured substance (eg sulphur) or just painted with paint from the same can.

  • $\begingroup$ He asked for exactly the same color, not just indistinguishable. $\endgroup$ – palsch Jun 12 '16 at 10:07
  • $\begingroup$ there's a difference? $\endgroup$ – Jasen Jun 13 '16 at 12:11
  • $\begingroup$ Yes, I think so. Indistinguishable is when you can't see the difference with your eyes, but the same color is when you have two objects that don't even have differences when you use very good machines. $\endgroup$ – palsch Jun 13 '16 at 13:49
  • $\begingroup$ indistinguishable applies to machines too, $\endgroup$ – Jasen Jun 13 '16 at 23:45
  • $\begingroup$ OK, so sorry, I'm no native speaker. +1 then. $\endgroup$ – palsch Jun 14 '16 at 4:27

My ideas to complete 5 tasks:

  • Square Wave
  • The equation of a general square wave is given by $S(x)=A\cdot (-1)^{\left\lfloor\frac{2(x-x_0)}{T}\right\rfloor}$. (See Wolfram Mathworld entry ) No one said I can't put $A=0$, thus attaining a square wave of $0$ amplitude.

    So to achieve this, simply

    Do nothing!

  • Exact circle
  • As suggested by humn, draw a circle of $0$ radius.

  • Endless Book
  • Make a book with $0$ pages. If you think it's not endless, tell me the last page number!

    Or if that sounds like cheating,

    manshu' s idea of spiral binding looks fine.

  • Annual apparatus
  • Take a disk with a small hole, and place it horizontally above a photographic plate. There will be a dot on the plate every noon, and the dot will be on the exact same position after a year.

  • Objects of same colour
  • Since the question asks to "Find" to objects, and to find usually means to get something which already existed, digital imagery doesn't seem acceptable to me. Instead, take the inner sides of your eye-lids. 'Colour' has meaning only when you see it, and to see them you need to close your eyes, and so they are both black.

    • $\begingroup$ Objects of same colour: Would an object and the objects reflection in a mirror be the same colour? $\endgroup$ – B540Glenn Jun 20 '16 at 14:51
    • $\begingroup$ What about the OP's stipulation: "I will not allow any NULL answers."? $\endgroup$ – Lawrence Aug 29 '16 at 15:06
    • $\begingroup$ @Lawrence It's ambiguous: I interpreted it to mean "I wouldn't allow any answer that does not include the solution to one or more of the tasks". $\endgroup$ – Ankoganit Aug 29 '16 at 15:19
    • $\begingroup$ @B540Glenn What if the mirror is tinted? $\endgroup$ – Ankoganit Aug 29 '16 at 15:20
    • $\begingroup$ I changed the wording of NULL answers to trivial answers. Think of it this way, I wouldn't count something as both an exact circle and an exact square. $\endgroup$ – Tony Ruth Sep 1 '16 at 22:45

    Excite me a wave which is exactly square. For the non-technical readers, waves come in four types: sine, square, triangle, and sawtooth.

    However, to quote Wikipedia, "An ideal mathematical square wave changes between the high and the low state instantaneously, and without under- or over-shooting. This is impossible to achieve in physical systems, as it would require infinite bandwidth." So, this one cannot be done.

    Draw me an exact circle.

    This one is fairly easy. Use a compass.

    Brighten my night with a light that is exactly white. This line could be interpreted in more than one way - light that is white (contains all parts of the EM spectrum) or that appears white (stimulates all of the cones in the eye).

    On my cursory reading I thought this would be impossible, as what is considered white can change depending on the context. But the question is asking for a white light, not a white object; thus, moonlight will do. (It is night, after all.)

    Show me a container that is exactly full

    All atoms contain lots of empty space between the nucleus and the electrons. Thus, a container full of water still is not full, nor is there any way to squash anything else inside. However, the OP [confirmed] that we're dealing with an HTML container. Nevertheless, there's still a problem: the definition of full is "completely filled," and you're always able to fit more code inside an HTML container. So I don't see how this one is possible. (You think I'm being picky? I just want things done exactly right.)

    Bind me a book which is exactly endless

    Spiral book. Take out the cover, and there's always another page if you keep flipping them around. Credit to manshu.

    Make me an apparatus which rotates exactly once per year Rotates in respect to what? The question asked that we "make" something,

    so can I make an anything, leave it exactly where it is, and let it sit for a year, during which time it will have rotated around the sun? (Which was said to be correct, but imprecisely worded. We are trying to be exact, aren't we? IMO a better answer would be to build a contraption like an egg timer or a watch whose hand/dial rotates exactly once per year.

    Find me two objects of exactly the same color

    As I mentioned in the white light line, our sense of color changes depending on where it is, what light is shining on it, etc. The only way two objects can be of the same color is if either they're digital, or if they can be confirmed to have the same amount of the same light waves bounced off of them no matter what is - a black hole. (Credit to humn.) Vantablack gets pretty darn close, but the question asked for it to be "exactly" the same color, and there's still that 0.035% of possibly different-colored light that bounces out.

    In conclusion

    Exact circle, exactly white light, exactly endless book, apparatus that rotates exactly once per year, and two objects with exactly the same color are exactly five items that can be performed exactly.

    • $\begingroup$ your answers are pretty good, and you've gotten very close to the spirit of this question. Here are my critiques: I'm not sure what you mean by use a compass. In regards to the container, I think the answer on whether or not it can be filled is dependant on what it contains. If it contains matter, sure you can keep adding to it, and in the html example, if you say its a container for code then its not full, but if its a container for an object and you put one object in it, then its full because you cannot insert more objects. $\endgroup$ – Tony Ruth Sep 1 '16 at 22:33
    • $\begingroup$ As to the rotating apparatus, consider this. If you put an object on the surface of the Earth, it will revolve around the earth every day, around the Sun every year, around the galaxy in whatever timespan that is, and it will pick up the Earth's precession. That's a lot more than 1 rotation per year, so it does not fulfill the requirements. Unlike translational motion, there are consequences of being inside a rotating system, like centripetal and coriolis forces. $\endgroup$ – Tony Ruth Sep 1 '16 at 22:37
    • $\begingroup$ @TonyRuth What's wrong with using a compass? Stick the one end in the center, drag the other end around in a perfect circle around it. And what was wrong with my egg timer? $\endgroup$ – user24580 Sep 2 '16 at 3:23

    Partial answer:

    Square wave

    "An ideal mathematical square wave changes between the high and the low state instantaneously, and without under- or over-shooting. This is impossible to achieve in physical systems, as it would require infinite bandwidth." - Wikipedia

    White light:

    white light
    Apparently colourless light, for example ordinary daylight. It contains all the wavelengths of the visible spectrum at equal intensity.
    Hmm, I should propose that I shine three red, green and blue lights all at the same time, with varying frequencies, and at one time the light must pass this point here,
    as defined by the Commission Internationale d'Eclairage (CIE).

    Draw me a circle which is exactly round:

    raʊnd/ adjective
    1. shaped like a circle or cylinder.
    Hence any circle is 'round'.
    Correction after OP's clarification: Suggested by humn
    Note, a circle is the set of all points in a plane that are at a given distance from a given point, the centre; equivalently it is the curve traced out by a point that moves so that its distance from a given point is constant. The distance between any of the points and the centre is called the radius.
    Hence it is nominated that I draw a circle of radius 0.

    Show me a container that is exactly full

    Full of air should do it, shouldn't it? Unless you want to argue that atoms actually have lots of space between the nucleus and electrons and therefore a container actually can't be full. Update (as shown in palsch's answer):
    Draw an object digitally

    Two objects of exactly the same colour:

    Cut one in half?
    Otherwise proposed by humn:
    Black hole (i.e. no colour)

    Proposed by manshu, for spiral book:

    "We can use spiral binding and take out the book cover. In this way, whenever we turn the page, the page will become the last page of the book."

    Hence the two that are not possible are:

    Square wave and yearly rotating apparatus.

    The five that are possible are described above.

    • $\begingroup$ air is compressible, when is it exactly full? $\endgroup$ – Jasen Jun 11 '16 at 8:17
    • $\begingroup$ ordinary daylight does not contain all visible frequencies in equal parts, this is how helium was discovered. you'll get white light from a 4700K furnace. $\endgroup$ – Jasen Jun 11 '16 at 8:31
    • $\begingroup$ I changed the circle prompt. Your container of air is not full since more air can be inserted. I'll count the cutting two objects in half, but I think there are also other ways to do it. $\endgroup$ – Tony Ruth Jun 11 '16 at 18:14
    • 5
      $\begingroup$ About the book....We can use spiral binding and take out the book cover. In this way, whenever we turn the page, the page will become the last page of the book. $\endgroup$ – manshu Jun 11 '16 at 18:25
    • 1
      $\begingroup$ A (mumble) might be the easiest-to-draw perfect circle. $\endgroup$ – humn Jun 11 '16 at 20:56

    Excite me a wave which is exactly square.

    Give me some graph paper and I'll draw a perfect square for you. If more precision needed, I can also make this graph digitally (For e.g. In Python using Matplotlib).

    Draw me an exact circle

    A point

    Brighten my night with a light that is exactly white.

    Nah, I can't wake you up. Seems impossible.

    Show me a container that is exactly full. (credits Palsch)

    <div id="container" style="padding:0;"> <div id="stuff" style="margin:0;width=100%;height=100%;"> Stuff </div> </div>

    Bind me a book which is exactly endless. (Reader will always read left to right, top to bottom, one page after another)

    I can do this digitally. Since you didn't specify anything about the reader going back, therefore whenever reader request a new page, I'll generate random text. Hence, whenever reader finishes a page and requests for the next one, he always gets another page.

    Make me an apparatus which rotates exactly once per year.

    I'll make some apparatus and then we have to go to one of the poles to check it's validity. I bet this'll rotate once a year.

    Find me two objects of exactly the same color. (credits Palsch)

    In JavaScript:
    var a = object1(); var b = object2(); a.color = "#f00"; b.color = "#f00"; //Given that both object1 and object 2 have an attribute named color


    Excite me a wave which is exactly square.

    A square wave basically plots between a low point and a high point, only - it is a non-sineoid waveform, according to wikipedia. So... binary? Plot your ones, and your zeroes, in a scatter-plot. Then, connect the dots - you know it has no in-betweens, so straight up-line form zero to one and straight down-line from one to zero, all on a forward line for time. If your computer is being all picky about lag-time and the time spent on rising and falling, tap it firmly and plot the silly thing by hand. A perfect square wave has an equal number of highs and lows (50% duty cycle) so make sure your binary message is long (or short) enough, or specially formulated enough, to get a perfect square wave. Or else "test plot" just alternating ones and zeroes.

    Draw me an exact circle.

    Take an eyedropper and a container of pure water. Carefully hold the eyedropper straight, and let a single drop of the water fall. While it is falling, its circumference at the equator should be an exact circle, since water's cohesion should pull it together in a sphere. Granted, it's hard to measure - but the circle has been created, and drawn with a description rather than a line. For an alternate method, build a space shuttle (or talk to those who have one) and repeat the experiment in space, where the shape of the water should form and exact sphere.

    Show me a container that is exactly full.

    Take a solution of distilled water and common dish soap, in a 6-to-1 ratio (optional: add glycerin or corn syrup). Mix well. Take a straw, a length of pipe, or a bubble wand, dip the end in your solution so that a thin film covers it, and gently blow through the hole to blow bubbles. They are a container, check. They are full (of air) also check. No more air can be placed in them, and none taken out of them, or they will pop.

    Or, alternately

    Take a sponge, and set it in a bowlful of water. Let rest until fully re-hydrated (some squeezing and expanding may help). The sponge is now exactly full of water. The fact it can be taken out of the water, and still be waterlogged, makes it a container. If you take it out of water, without squeezing, it may still be full of water (perhaps exactly so for a little bit) - but it's harder to judge the "exactness" of the fill or exactly when it is done dripping.

    Bind me a book which is exactly endless. (Reader will always read left to right, top to bottom, one page after another)

    Start with a standard hole-punch, and the correct number of binder rings (most hole-punchers make three holes, some do four). Binder rings are effectively smooth (though ones with as little difference in the hinge and clasp are preferred), so any pages, when turned, will quickly circle around to land at the back of the sheaf of papers. To make "truly" endless, add increasing numbers of pages until they have comfortably filled the ring, and the book can't "close" to a single sheaf but must remain standing, with the pages equally splayed out around the circumference - it can still be read, since even when full at the ring site, the edges are quite loose and can be turned. To make it a book, fill the pages with something, perhaps art and complex images, so one can continue revisiting the pages indefinitely, rather than a simple story which might not make sense out of order or may not be worth rereading.

    Make me an apparatus which rotates exactly once per year.

    Making a clock-like mechanism shouldn't be too hard. Make sure it's sturdy enough to last at least a year (under any conditions), and also - instead of rotating a hand, have the motor rotate a disk set atop another disk, just for aesthetics. Have on top of the second disk, another rotating mechanism that can be set at an angle. Take it and run away - north, or south, depending on your location and travel habits, but you need to get over one of the poles - and I mean right over the turning point, not the magnetic pole or what have you. Set your apparatus down right over the pole (consider setting up a shelter over it, you built it study but why take chances), and set the mechanism to turn once per day - clockwise, exactly one rotation per 23 hours, 56 minutes, and four seconds. The earth rotates counterclockwise at the same rate, bringing the rotating disk of the apparatus to a halt regarding the earth's daily rotation. The second mechanism should be set to one rotation per year, and the mechanism tilted by 23.4 degrees - this should counterbalance the earth's tilt. In exactly one year, though, it has rotated exactly once around the sun - the rotation will occur at exactly the one year mark because one rotation about the sun is the actual definition of a year, it is 365:5:48:45 on average. Larger rotations, like the sun's or galaxy's rotating, are so immeasurably large, and the portion our apparatus got through in a year so small, that they don't count - that is, exactly one sun-rotation, and some extra wobbly non-rotation distance moved.


    addition: Because the axial tilt is a genuine pain, take our clock-mechanism, head over to the pole, and build a tiny railroad in a complete circle around the pole. It needs to be exactly 1,618.60075 miles long, which means 257.6085 miles from the pole in every direction (you may want the south pole for the extra room). Set your device on a tiny train on the tiny railroad, set it going counterclockwise with a speed of 3.25 inches per second - which will really cancel the axial tilt, and prevent a small second rotation from being traced by the device spinning around the tilt-axis.


    go back to the folk you got your space ship from back in drawing an exact circle, and borrow a space ship for a brief jaunt over one of the poles (and their computers) and plant the device...in space, in a reverse-geosynchronous orbit, directly over the axis perpendicular to the earth's rotation (that means, 23.4 degrees from the pole) or else in a sun-synchronous orbit over the equator. Use the two rotating mechanisms to cancel out any other rotations you would like to eliminate (the first will cancel the tilt but the daily rotation will need countering, the second will cancel the daily rotation but the tilt will need countering, and one to spare if the satellite it's attached to might rotate and that needs countering, whatever).

    Find me two objects of exactly the same color.

    Take two samples of a pure substance - composed of a single element or reliably pure compound. It is important that the substance be pure, 100%, but not really what the substance is - even distilled water will do as long as it is pure. Since we want two objects, you could even purify the samples from different batches - and for potential bonus points, have them identically shaped or held. Take the samples into a lead-lined darkroom, set them on a table, and shut off the lights. Your objects are now exactly the same color. Since they are a pure substance, no chemical test can distinguish between the wavelengths of light they will reflect, and in such perfect darkness, your eyes (or anyone else's) can't be fooled by apparent differences brought on by exact shape, any containers, or specific placement in regards to the light source (angles or shadows).


    Find me two objects of exactly the same color.

    The quantum chromodynamics tells us that quarks hold one of exactly three color charges. So you can just take four random quarks and two of them are bound to be exactly same color.

    Draw me an exact circle.

    You will need a steel ball, a smooth level surface, a nail and a rope. You take a steel ball and attach the rope to it. Drive the nail into the surface, and attach the other end of the rope. Now spin the ball with enough velocity such that the rope stays tight. The ball will make an exact circle.

    Or easier:

    Put this into postscript file: '0 0 3 0 360 arc closepath stroke'.


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