## Thursday, May 31, 2012

### NPR Sunday Puzzle (May 27, 2012): Types of Wool - Actor Puzzle

NPR Sunday Puzzle (May 27, 2012): Types of Wool - Actor Puzzle:
Q: Name two different kinds of wool. Take the first five letters of one, followed by the last three letters of the other. The result will spell the first and last name of a famous actor. Who is it?
Take the actor's first name and add a type of idol. Anagram the letters to get the actor's first movie.

Edit: Al + Matinee = Me, Natalie
A: Alpac(a) + (Mer)ino = Al Pacino

## Saturday, May 26, 2012

That Darn Achilles:
Q: You, Paris, have the luxury of launching an arrow at faraway Achilles either from the ramparts above the Hesperian Gate at a height of exactly 8 meters. Or you can stand atop Priam’s palace. This gains you another 7 meters of launch height, but it costs you 15 meters of horizontal distance. If the arrow leaves your bow at a somewhat modest 70 meters per second, are you best taking your pot-shot at far-off Achilles from the ramparts or the palace? Which perch offers the farthest reach?

I had to look up the formulas for determining the maximum range of a projectile when fired on uneven ground. The following page was invaluable.
Wikipedia: Range of a Projectile

Because several things weren't stated, I'm going to assume that we can use acceleration of gravity (g) at sea level, we can assume no wind resistance and also assume that the ground is level between the target and the firing point (except for the elevation change and horizontal offset provided by the ramparts (0,8) and the palace (-15, 15).

While the ideal case (firing on even ground) results in an optimal angle of 45°, when you are firing from a height, then you want to angle down slightly to maximize distance. I won't bore you with going through the details on that page, but basically there's an equation for the horizontal and vertical positions at time t, given an initial angle (theta) and velocity v. You can then set the final height to be 0 and solve for t. Using that you can get an equation for distance given an angle and by taking the derivative and setting it to zero, you can get a formula for the optimal angle to get the longest distance.

Rampart (x0,y0) = (0,8)
Palace (x0,y0) = (-15,15)
Velocity (v) = 70 m/s
Gravity (g) = 9.80665 m/s^2

Optimal angle (θ) = cos-1 [ √(2*g*y0 + v^2) / (2*g*y0 + 2v^2) ]

Rampart angle (θ) = 0.777519 Radians or 44.54853°
Palace angle (θ) = 0.7708234 Radians or 44.164927°
Distance (d) = [ v*cos θ [v*sin θ + √((v*sin θ)^2 + 2*g*y0) ] /g + x0

Rampart distance = 507.5979m
Palace distance = 499.44231m

A: When firing from the ramparts (8m), the optimal angle is around 44.55° and will net you a distance of 507.6 meters.

When firing from the palace (15m), the optimal angle is around 44.16° but because of the -15m offset you only reach 499.4 meters.

## Thursday, May 24, 2012

### NPR Sunday Puzzle (May 20, 2012): Present and Past Tense Verbs

NPR Sunday Puzzle (May 20, 2012): Present and Past Tense Verbs:
Q: Think of a common three-letter word and five-letter word that together consist of eight different letters of the alphabet. Put the same pair of letters in front of each of these words, and you will have the present and past tense forms of the same verb. What words are these?
I appear to be drawing a blank...

Edit: My hints were "drawing" (with ink) and "blank" (as in zero/ought). In the comments I had other hints like "I'm not kidding" (i.n.k.), "last decade" (the oughts), "Oops, unfortunately good hints take..." (o.u.g.h.t.), "I now know" (i.n.k.), "I figured" (thought) and "just assume" (think).
A: INK & OUGHT --> THINK & THOUGHT

## Thursday, May 17, 2012

### NPR Sunday Puzzle (May 13, 2012): Capital Profession

NPR Sunday Puzzle (May 13, 2012): Capital Profession:
Q: Name a state capital. Change one of the vowels to another vowel and say the result phonetically. You will name a revered profession. What is it?
While I initially was down with a different answer, I realized that Will included the word phonetically for a reason.

Edit: My hint was "While I initially" which means take the initial letters of "While I" (WI) to get the state. The different answer that some came up with is Dover/Diver, but it isn't necessary to pronounce diver phonetically, so that isn't the intended answer.
A: Madison (Wisconsin) --> "Medison" = Medicine

## Sunday, May 13, 2012

Coffee Conundrum II:
Q: Each cup of coffee gives me a jolt and then the jolt decays across time according to the following equation (t in minutes): Jitters=10-[(t-10)^2]/10
So at minute 10 after consuming a cup of coffee (which, for the purposes of this puzzle happens instantly), I reach a maximum of 10 jitters. At a combined 20 jitters, I go catatonic. With what frequency can I instantly consume coffee without the combined jitters passing this important tipping point?
The diagram shows the exact tipping point situation. Focus on the red curve which is the second cup of coffee. Right at the middle of the parabola, the second cup is at its maximum (10 jitters). At this point, the effects of the first cup are diminishing and the effects of the third cup are increasing. Notice, because of symmetry, they have the exact same value when they cross. The tipping point will be when cup 1 is contributing 5 jitters, cup 2 is at its maximum of 10 jitters and cup 3 is also contributing 5 jitters (5+10+5 = 20). (Note you can extend this diagram out to 4 cups, 5 cups, etc. but you'll never have more than 3 cups contributing to the total jitters at once. If you did, you'd definitely be over the tipping point of 20.)

As stated in the puzzle, the height of one parabolic curve is given by the formula:
Jitters = 10 - [ ( t - 10 ) ^ 2 ] / 10

Solving this for a Jitters value of 5 we have:
5 = 10 - [ ( t - 10 ) ^ 2] / 10
-5 = -[ ( t - 10 ) ^ 2] / 10
5 = [ ( t - 10 ) ^ 2] / 10
50 = ( t - 10 )^2
t - 10 = √50
t = 10 ± √50
t = 10 ± 5√2

That means that the coffee Jitters are at a height of 5 either 5√2 seconds before the peak or 5√2 seconds after the peak. This also happens to be the minimum frequency between cups (5√2 seconds or approximately 7.071068 seconds apart) to avoid jitters.

If you can handle exactly 20 jitters without going catatonic, then you could handle drinking cups of coffee every 5√2 seconds (≈7.071068 seconds)

If you must stay under 20 jitters, then you would have to pick a frequency just over every 5√2 seconds.
A: Frequency of 5√2 seconds (or approximately every 7.0710678118654752440084436210484903928483593768847403 seconds)

## Thursday, May 10, 2012

### NPR Sunday Puzzle (May 6, 2012): Bronte Sisters Turn a Phrase

NPR Sunday Puzzle (May 6, 2012): Bronte Sisters Turn a Phrase:
Q: Using only the six letters of the name "Bronte," repeating them as often as necessary, spell a familiar six-word phrase. What is it?
The Bronte Sisters grew up in a small village called Haworth. The question is whether this is relevant to the puzzle.

Edit: A small village is a hamlet and the famous soliloquy continues with "...that is the question".
A: To be, or not to be

## Thursday, May 03, 2012

### NPR Sunday Puzzle (Apr 29, 2012): Capital Punishment

NPR Sunday Puzzle (Apr 29, 2012): Capital Punishment:
Q: Name the capital of a country that, when said out loud, sounds like a three-word phrase. This phrase might describe the reason why the police did not catch a barefoot thief. What is the capital, and what is the reason?
We seem to have a pretty characteristic Will Shortz puzzle involving countries, phrases and sounds.

My clue was "characteristic" which contains the letters of the country name in order (cHarActerIsTIc). I'm not going to get into a debate on the French vs. anglicized pronunciation of the capital city. Will must have heard it pronounced "port-oh-prints" just like I have. Get it? The police weren't able to identify the barefoot thief because all they had were "poor toe prints."
A: Port-au-prince & "Poor toe prints"