bonanova

Bravo! Best answer so far. Can you generalize for other ranges of x?

Both statements are true. But you can arrange the rationals into a square array [number the columns and rows with integers and the elements of the array are their quotient] and "count" them, following an expanding diagonal path through the array. So there is a 1-1 correspondence between the natural numbers 1, 2, 3 ... and the rationals p/q. You cannot so arrange the reals. Between any two entries of the discrete array of rationals there is an infinity of reals. You can't squeeze more rationals into the table - they're all there. We say that countable infinite sets [like the natural and the rational numbers] have a certain cardinality [Aleph0] The cardinality of the reals is a higher cardinality.

Between any two rational numbers, no matter how nearly equal they are, there is an infinity of irrational numbers. Given that the irrationals "outnumber" [a poor choice of word, since they in fact are not countable at all; the rationals are countable] the rationals so greatly, a real number chosen at random has zero probability of being rational. Zero probability does not mean impossible. Well, it's a mathematical thing. Any physical cut will disturb the pie. But you can mathematically "cut" the continuum of real angles at an irrational value, yes. It's precisely these "cuts" that are used to define the real numbers. [Google "Dedekind cut" or just look here.]

This one has no time limit... Four gold stars. Great job.

Dawg has it.

So it's your birthday! And the round thing with candles on it is from Carvel's. You blow out the candles in one try, and your wish comes true! A personalized BrainDen puzzle - which, unfortunately, you must answer correctly before eating the sweet cold stuff. It's French vanilla ice cream, covered with dark Swiss chocolate icing. A protractor is provided, which is set at random to an angle X. You must cut a piece at that angle, turn the piece upside down and replace it. Because it's made of ice cream, it heals itself [re-freezes]. You then cut a 2nd piece, at angle X beginning where the 1st piece ended, and turn it upside down. Beginning where the 2nd piece ends, a 3rd piece at angle X is cut and flipped. And so on. Each successive piece begins at angle [n-1]X and ends at angle nX, [shown] leaving a path of vanilla ice cream in its wake. Eventually you will work your way around the cake and begin cutting into the part of the cake that has been flipped upside down. When that is flipped, of course, the chocolate icing comes back on top. Here's the puzzle. If the cut-and-flip-at-angle-X process is continued, will there ever come a point where the cake is restored to its original state? Answer correctly, and we all have cake. Otherwise, it's wait until next year.

Just need QE2's diesel mileage. Make a guess; I'm giving you 2 orders of magnitude leeway both directions . For example, you guess 1 mile per gallon. If it's anywhere between 100 miles per gallon and 1/100 miles per gallon, you win! Be the first!

Used in normal conversation. Not uncommon.

Thank you. With regard to my answer to the OP I declare truce. I did not say the moon would be as directly informative as looking at the daylight sun. Only what you say; which is all that I needed to answer the OP. ------------------ The "at the moon's zenith" statement is still not clear to me. Unless it also needs to be adjusted for season, time, latitude, phase, kitchen sink, what have you: For one thing, the full moon is a degenerate case, not pointing left or right; OK no problem with that. From one day before full to one day after full, it switches exactly 180 degrees, at all latitudes? Or just at the equator, ignoring the ecliptic, or ... ? Here are some data: This morning [41o N latitude], 4 or 5 days after full moon: At 3:00am, prior to zenith, the waning gibbous moon pointed downward, to about "7:00" as shown in my 3:00 am sketch. At 6:50 am, after zenith, the sun being just below the horizon, it still points somewhat downward, as in my 6:00 am sketch. At zenith, therefore, [perhaps 4:00am], the sun being further below the horizon than now, La Luna also must have pointed downward. [Note added at 9:30 am a couple hours after sunrise] Moon now points to about 10:00 on its face, in agreement with my 9:00am waning gibbous sketch. [/Note] My sketch should show zenith somewhere between moons 3 and 4 - not at moon 3. Also my 1Q sketch should mirror the 3Q sketch. For waxing crescent and gibbous phases, the sun is pre- and post- setting, at moon zenith; so the light/dark boundary isn't vertical. [my waxing crescent sketch should be changed in that regard] For waning crescent and gibbous phases, the sun is pre- and post- sunrise, and the same conclusions obtain. At 1Q and 3Q, the moon's zenith is at sunset and sunrise, creating a vertical light/dark boundary. When I was an experimental scientist I loved theories and I loved data. When they didn't agree I always sided with the data.

Trivia quiz. Try these without using computer aid. On a standard typewriter keyboard, what is the longest English word in common use that can be typed using keys on the left side [q w e r t a s d f g z x c v b] * answered on the right side [y u i o p h j k l n m] * answered on one row [q w e r t y u i o p], [a s d f g h j k l], or [z x c v b n m] * answered What English word in common use ends with the letters 'mt'?* answered What one-syllable English word in common use that is not the name of a color [e.g. orange, purple, silver] has no rhyming word?* answered What four English words in common use end in 'dous'? Do not count various forms.* answered What animal's brain is smaller than its eye? [be nice] <_< * answered For a month not to have a full moon, it must be shorter than 30 days, and therefore be February. This has happened only once in recorded history. To within 100 years, what year? * disputed but answered Name an animal that has become domesticated in the last 4,000 years. [be nice].* answered If the population of China were to walk past you 8 abreast, what time interval to within a factor of 10 would elapse, before the trek would be completed?* answered In this age of fuel economy, to within a factor of 100 how many miles per gallon [diesel] at cruising speed does the cruise liner QE2 get? * answered Does the world's population [of living people] exceed the number of living chickens? * answered Who blink more frequently, men or women? * answered

Yeah.

These are all correct. And of course you can add the leading number in the bottom row of repeating digits.

Here's a little party trick. A friend silently chooses three consecutive numbers [say 49, 50, 51] none greater than 60. He then announces a multiple of 3 [say 54] that is less than 100. Finally he multiplies their sum [204] by 67 [13668] and announces the last two digits [68]. You pause for a moment to think, then you tell him his starting number [49] and the remaining digits [1, 3, and 6]. Summarizing: he tells you 54 and 68; you tell him 49, 1, 3, and 6. Wow, he gasps. How did you do that? A real magician never tells. But you can tell us, in a spoiler.

As you word it, N appears to have 3 as upper limit; and that has been solved. One might ask for N>3, but physical matches can't supply a verification.

Would a hint be useful?

Well, so far the moon hasn't taken issue with my sketches. If the sky is within walking distance, one can check that at 1:45 am ET it looks very like my 3:00 am waning Gibbous. And in a couple weeks, should anyone still care, more of the various theories can be checked out. Sketches aside, the point remains: the lighted pattern on the face of the moon points toward the sun. That's relevant to the OP because, season and latitude taken properly into account, information regarding the compass points can be drawn from the position of the sun. That position can come either directly, from observation during daylight; or indirectly, by observation during moonlight. I'm happy to stay with that. Truce.

We have a waning Gibbous moon - 3 days past full. At 22:00 EST last night the moon was in the eastern trees; its dark side was seen to be about 1:00 on the moon's face - pointing down left to about 7:00. The pattern continued to rotate clockwise, as my sketch shows. I photographed the moon between 04:30 and now, 08:45 EST, when it set into the western trees, In that interval the "pointing" area rotated between 9:00 and 10:00 on the moon's face [see the morning waning Gibbous sketches]. The moon pointed east after midnight. If skies are clear tonight I'll try to photograph the earlier orientation.

Not that it was quick to do, but if you decide to make a correct version of my sketch, I'd be interested to see it. I was thinking about 45o N latitude, but equator is fine - first-order sketch is good enough.

If I arrange four matches perpendicularly end to end, [forming a square] I make four right angles. If I move parallel matches inward, pairwise, [making a # figure - tic-tac-toe] I make sixteen right angles. How many right angles can be made from just three matches?

Time travel extraordinaire ...