Monday, December 10, 2007

Monty Hall Problem Revisited

After attempting to answer someone's supposed counterargument in the comments section to my earlier post, I realized that there's a better explanation of the Monty Hall Problem that will hopefully satisfy/silence the doubters. First, I'll restate the problem:

Monty Hall Problem

Of three doors, one is good. First, you select a door. Second, Monty Hall eliminates one of the doors for you - the door he eliminates is not the door you selected, and not the good door. Third, you have the option of keeping your original door, or changing to the only remaining door. What should you do?

SOLUTION

For the sake of this explanation, imagine that you play the game many times, and you always pick Door A, which means Monty will always have to eliminate either Door B or Door C. Every other aspect of the game is still fair and random, meaning that at the beginning of each game, Door A, Door B and Door C have an equal probability of being good.
Now let's consider the moment after you've selected Door A (because you always select Door A in this example) and Monty must eliminate Door B or Door C for you.
  1. If Door B is good, he will eliminate Door C (1/3 of all occurences)
  2. If Door C is good, he will eliminate Door B (1/3 of all occurences)
  3. If Door A is good, Monty must choose whether to eliminate Door B or Door C. He does so with equal probability, so:
  • If Door A is good, sometimes Monty eliminates Door C (1/6 of all occurences)
  • If Door A is good, sometimes Monty eliminates Door B (1/6 of all occurences)

In 1/2 of all the games you play, Monty eliminates Door C (1/2 = 1/3 + 1/6) and you must then choose between Door A and Door B:
That means that for every 6 times you play the game, you'll have to choose between Door A and Door B 3 times. Of those 3 times, 2 of them will occur because Door B was the good door, and 1 of them will occur because Door A was the good door and Monty randomly eliminated Door C.

And the other 1/2 of the time (3 of every 6 games) you find yourself left with a choice between Door A and Door C, and it's twice as likely that Door C is the good door:
In either event, your chances are better if you change your original selection.

Sunday, December 9, 2007

Superman Depantsed

My favorite free video podcast on iTunes is the Vintage ToonCast, which features an assortment of excellent old cartoons. Among them are 13 of the 17 original Superman cartoons made by Fleischer studios (Popeye, Betty Boop) in the early 1940s. Some stray thoughts/observations/reasons why you should watch them:

  • The narrative is as reliably formulaic as later Scooby Doo cartoons, if not exactly as fill-in-the-blank simple: The villains announce/begin their dastardly activities, Clark and Lois investigate, Lois manages to get into harm's way (often as she edges out Clark for access to the story), Clark sneaks away and Superman appears, he struggles briefly but ultimately defeats the villains with ease, Lois and Clark share brief dialogue that demonstrates Lois is still unaware of Clark's other identity.
  • This is all punctuated by the oft-repeated theme music, which has the effect of emphasizing how quickly everything gets resolved in the 8-minute episodes, reassuring you that everything is very alright, or will be alright again very soon. It plays over the opening credits, again 30 seconds later when Superman's character is introduced at the end of the prologue, again around the 5 minute mark when Superman first appears, occasionally repeats as he strikes successful blows against the villains, again as he finally defeats the villains, and continues through the epilogue to the end of the episode.
  • Dialogue takes a back seat to the very impressive artwork in these shorts. After the expository first act, the rest of the story is told almost entirely in pictures, and all human movements are animated realistically and without exaggeration, (using Fleischer's technique of rotoscoping). My favorite example of this is when we see Superman's shadow or silhuette removing his Clark clothes, as shown in the screen capture above, and often bending over and raising his knees as he takes off his pants. Here are six more such screen captures:
  • What little dialogue there is is also unexaggerated. I love the Clark actor's flat reading of the line "This is/looks like a job for Superman". At least once he adds a slight hint of emphasis to the word "is".
  • I'm no comic book expert, and I don't know in which of the many Superman incarnations they first suggested any romantic tension between Lois and Clark, but the only tension here is professional, as Lois selfishlessly works to gain sole access to each story, shutting out Clark in the process.
  • One of the themes among the cartoons available on this podcast is the surprisingly insensitive/offensive racial caricatures that were common in eras past, responsible for most if not all of the episodes' "EXPLICIT" tags. Several of the Superman cartoons pit our hero against WWII enemies (and one African tribal cult under Nazi control), but the artwork, voices and plotlines are extremely tame and respectful when compared to some of the cartoons featuring Bugs Bunny and others.

I also highly recommend several of the other cartoons on the Vintage ToonCast. I haven't watched all of them yet, but for now I'll suggest these:
  • All of the Betty Boop cartoons (#5-10, some unnumbered). I'd never seen one before and was pleasantly surprised, especially by some of the trippy graphics in Snow White (#6) and Minnie the Moocher (#10). Watch them for the Cab Calloway music alone.
  • The famed "Duck and Cover" video (#36).
  • Any or all of the "Private SNAFU" cartoons (#26-28, some unnumbered), depicting a foolhardy soldier, voiced by Mel Blanc (Bugs Bunny), who demonstrates how not to serve your country.
  • "Mr. Finley's Feelings" (#54). Cartoon about a person with rage issues done in an interesting tone. Note the use of 1st person perspective, and see if you can spot at least one curious moment (not the flashback) where they violate it.
  • "The Trip" (#51). Happy bicentennial America! Let's celebrate with eye candy for the drug enthusiast!
  • "Going Places" (#57). Explaining and championing the profit motive (lest you find yourself tempted by communism).
  • "The Early Worm Gets the Bird" (#16). Not necessarily the most offensive of the episodes featured on this podcast, but definitely the most thorough exploration of a racial caricature.
  • "Fresh Hare" (#1). A surprising and absolutely unnecessary ending to a Bugs Bunny toon.

Monday, November 26, 2007

Algebra Puzzles!

As if the word "algebra" wasn't off-putting enough, my first two problems are normally formed using trains, evoking the stereotypical headache-inducing algebra problem ("One train leaves Tulsa going northeast at 60mph, the other..."). I've replaced the trains in an attempt to make the problems more attractive to the few readers I haven't already alienated with this series of posts.

Man in a Tunnel

A man has traveled 1/4 of the way through a tunnel when he notices a giant killer bunny hopping at a constant rate behind him, some distance outside of the tunnel. Whether he continues forward as fast as he can, or turns around and runs as fast as he can, he will exit the tunnel and miss being devoured by the bunny with zero time to spare at either end. How much faster is the mutant bunny than the man?

Supersonic Bee

Two giant killer bunnies, initially separated by a distance of 25 miles, are both hopping at a rate of 100mph, directly towards one another. A supersonic bee begins on one bunny's nose, and flies back and forth between that bunny's nose and the other's at a rate of 800 miles an hour until the bunnies ultimately collide with each other and the bee. At the point of impact, how far will the bee have traveled?

Heaven and Hell

A giant killer bunny dies and finds itself given three unmarked doors to choose from. One leads directly to heaven, one results in a 1-day stay in hell, and one results in a 2-day stay in hell. If the bunny chooses either of the "hell" doors, it will carry out its hell sentence and return to the same three doors, which at that point will have been randomly shuffled, and the bunny gets to choose again. What is the average time it takes a giant killer bunny to get into heaven?

Man in a Tunnel
SOLUTION

I find this problem interesting because it feels as though you're not given enough information to solve it. How fast can he run? How far away is the bunny?

More data would only be helpful insofar as it would make the problem easier to visualize than by using algebraic variables. So let's say it takes the man 1 minute to reach the end of the tunnel through which he entered. At this rate, it would take him 3 minutes to reach the far end of the tunnel, since it's three times as far. Adding the times for these segments together means it would take a total of 4 minutes for this man to run from one end to the other. How long would it take the bunny to cover the same distance? It would take the bunny 1 minute to reach the near end and 3 minutes to reach the far end, so 2 minutes to cover the entire distance of the tunnel. The bunny is twice as fast as the man.

Supersonic Bee
SOLUTION

Solving this problem hinges on realizing how simple it really is. If it actually required adding together all the successively decreasing distances between the bunnies, solving it would not be fun or interesting, and even most nerdy math types wouldn't bother.

All you need to know is speed = distance / time (or distance = speed x time). The bee's speed is 800mph. How long will he be flying? Traveling toward each other at 100mph, the bunnies can each cover 12.5 miles in 1/8 of an hour. So the bee travels at 800mph for 1/8 of an hour: 100 miles.

Heaven and Hell
SOLUTION

Solving this by algebra is more or less straightforward, but it's still very easy to get lost.

Call the answer - the average time it takes a bunny to get into heaven - X. X is the average of three values: the expected times for each door. Door #1 has an expected time of 0. But how can we calculate the others?

If the bunny walks through door #2, it spends 1 day in hell and starts over again. After that 1 day, it can then expect to wait the average time, X. So the total expected time for door #2 is 1 + X. The same reasoning gets you 2 + X for door #3.

Since X is the average of the three values, you need to add those values and divide by 3. Putting all this together you get the equation:

X = ( [0] + [1+X] + [ 2+X] ) /3

Some simple equation manipulation gets you 3X = 3 + 2X and finally X=3. The average time it takes a giant killer bunny to get into heaven is 3 days.

Monday, November 19, 2007

More Puzzles

Back by popular demand no demand whatsoever:

The Monty Hall Problem

Three Doors. One of them is good (it leads to underwear models, funnel cake, etc.) and two are bad (alligators, doctoral theses). You pick one, but before we show you whether the door you picked is good or bad, we generously eliminate one of the bad doors from the other two. Now it's time to make your final decision: Switch to the only remaining door, or stay with your original pick. Should you switch or stay? Does it matter?

Coin Flip Game

You and I flip a coin at the same time, and continue flipping at the same pace until the game is over. You win as soon as you get two heads on consecutive flips. I win as soon as I get a head and a tail, in that order, on consecutive flips. (We tie if both things happen on the same flip.) Who, if anyone, has better odds?


Unlike last week, intuition initially suggests simple and uninteresting answers to both puzzles - "it doesn't matter which door" and "no one has better odds", respectively. Of course, those intuitive answers are not correct.

The Monty Hall Problem
SOLUTION

The intuitive (but wrong) answer, "it doesn't matter", appears to make sense because (1) there are now just as many good doors as bad doors, and (2) you still don't know if your original pick was good or bad. (1) and (2) are both correct assumptions, but the actual answer is that if you switch doors, you'll pick the good door twice as often.

Think of it this way: When you originally picked your door, you had a 1/3 chance of being right. Nothing has changed that fact. When I eliminated a bad door, I was able to do so independent of whether your original pick was good or not. So now there's still a 1/3 chance that your original door is good, and a 2/3 chance that the good door is somewhere else. But "somewhere else" now consists of one option instead of two.

Coin Flip Game
SOLUTION

The intuitive answer is that the game is fair, because in any two consecutive flips, we both have a 25% chance of success, right? In reality, my odds of winning on a given flip eventually tend toward 50%, and your odds of winning on a given flip tend toward just over 19%. The mathematics turns out to be surprisingly complicated, involving patterns related to the Fibonacci sequence and the golden ratio. Figure that out on your own if you're so inclined. If you're not, here's a partial explanation of why the game isn't fair.

What are the chances of success on a given round? If your last flip was heads (good news for you), it's 50%. If your last flip was tails (bad news for you), it's 0%. Same for me. This logic holds as far as the second round. But if it's a later round and we're still playing, that means that nobody won on the previous round, and we can deduce a little more:

You didn't win on the previous round, and you needed HH, so your previous two flips could have been HT, TH, or TT. That's: one scenario in which you might win on your next flip, and two in which you can't.

I didn't win on the previous round, and I needed TH, so my previous two flips could have been HH, TH, or TT. That's two scenarios in which I might win on my next flip, and one in which I can't.

When you consider more and more previous flips, it gets increasingly complicated and the odds skew more and more in my (HT) favor. Here is a partial table of the odds of getting the second of two desired flips on a given round.

For example, if we make it to the 6th round of flipping (meaning you haven't yet made two consecutive heads) then there is a 38.4% chance your most recent (5th) flip was heads, and thus a 19.2% chance you'll get your second consecutive head on that 6th flip.

Monday, November 12, 2007

Puzzles

A while ago I came across a website purporting to list Google job interview questions, all of the brain teaser variety. I thought some of them were unlikely to appear in an actual interview as they've been kicking around for quite a while, but it revived in me a fascination with puzzles and brain teasers, and got me thinking about the different types and qualities of puzzles there are. I've decided to make something of a Monday tradition of presenting different puzzles and their solutions, for at least the next several Mondays.

Boy Country

In a country where people only want boys, every family continues to have children until they have a boy. If they have a girl, they have another child. If they have a boy, they stop. What is the proportion of boys to girls in the country?

Crazy Guy on a Plane

100 airline passengers are waiting to board a plane. They each hold a ticket to one of the 100 seats on the flight. (It doesn't matter, but imagine the first person has ticket #1, and so on.) The first person in line is crazy, and will select a seat at random, possibly even his own. Every person after him will sit in their own seat if it's available, and pick a random one if it isn't. You are the 100th person. What are your chances of getting your own seat?


These puzzles are both fine examples of what puzzles should be, in my opinion, because they fit three important criteria:
  • The solution is interesting, because it's not what you expected and/or simpler than you expected
  • Finding the solution requires only reasoning ability, not knowledge of higher mathematics
  • It's not "tricky" in the sense that the solver has to make any less-than-obvious assumptions ("you can melt the ice", "turn the sweater inside out", etc.)
Boy Country
SOLUTION


Using simple reasoning and no mathematics at all, you should be able to convince yourself that this birthing strategy does nothing to alter the natural gender ratio of 50/50 (or 51/49, according to some sources). Families have children. Some are girls, some are boys. Some families continue having children, some don't. As more babies are born, some are girls, some are boys. And so on.

If you're not convinced, consider 64 families. On average, 32 firstborns are boys, and 32 firstborns are girls. Some nine months or more later, 32 of those families give birth again: 16 have boys, 16 have girls. Next round: 8 boys, 8 girls, etc. The mathematics gets a little sticky when you get down to that last baby, but by then you have 63 boys and girls each to back up the 50/50 argument, and on a larger scale it's even more convincing.

Crazy Guy on a Plane
SOLUTION


For any problem involving an arbitrarily large number, it's always best to imagine it on a small scale first and see if any predictable patterns emerge. If it's just you and the crazy guy occupying 2 seats in a Cessna, your chances are 50%. Add one more person. Now the crazy guy has three options: his own seat ("good"), yours ("bad"), or the other person's seat. If he picks the other person's, then that other person has two options: the crazy guy's seat ("good") and yours ("bad"). Still 50%. Now consider the 100 person version. Crazy guy has one "good", one "bad" and 98 other options. If he sits in, say, passenger 57's seat, then passengers 2-56 get their own seats, and #57 has one "good", one "bad", and 42 other options. Any way you cut it, the odds will be 50%.

Friday, November 9, 2007

Main Event at Tokyo DisneySea

Yes, that's DisneySea, right across the street from Tokyo Disneyland. Every Disney park features a nightly show that can be seen from a large area of the park. I saw the one in Anaheim in 2005 and was quite impressed, but I'd say this one was as good or better. After a brief introduction by a Japanese-speaking Mickey, the show presented an embodiment of the Spirit of Water, then Fire, followed by something of a mating ritual between the two. In addition to the spectacle, I just appreciate the difference in storytelling style, focusing on harmony in nature, rather than following the kind of "hero vanquishes villain" narrative I'm used to.

Video 1 (1:11)
The Spirit of Fire, a metallic and tentacled seabeast that emits flames from its entire structure.


Video 2 (1:05)
Fire and Water, "flirting".


Video 3 (:54)
The marriage of Fire and Water.

Tuesday, October 30, 2007

A Review of Weapons and Accessories from The Legend of Zelda

Wooden Sword - I like how Link starts his mission without any weapons at all, and has to make an effort, albeit extremely minimal, to acquire his first item. I can't help wondering, though: Wouldn't it be more interesting if he had to navigate his way past a few enemies without a sword, or even his first shield? Additionally, the sword's throwing function gives Link a great incentive to maintain full life energy at all costs.

White Sword - This comes a little too early in the game. At the very latest, Link acquires it after completing level 2, so only foolish players who enter level 3 without it get to see the blob-like creatures split into chocolate chip-like creatures. An ambitious Link can get the white sword before even level 1, making the first two labyrinths a little less than challenging.

Magic Sword - Appearing halfway to late in the game, this weapon is one of the best payoffs. It's significantly more costly than its immediate predecessor, requiring you to complete the first four levels and find all five auxiliary heart containers (or perhaps more levels and fewer containers for the player who has yet to learn all the game's secrets). It's well worth the wait, and after finally acquiring it, Link gets to use it in up to five more levels. And unlike the flawed timing of the white sword, the magic sword's heart container requirements mean that you must first deal with the large jumping bat creatures splitting into smaller bats before later appreciating the convenience of slaying them with a single thrust.

Magic Shield - The difference between the defensive utilities of the regular and magic shields is just right. The fact that the magic shield can be involuntarily taken away from Link should he find himself trapped inside the layer-cake looking creatures who first appear in level 6 gives those otherwise unimposing enemies an added terror level, as Link has grown to appreciate the increased protection his magic shield provides him.

Bomb - This weapon proves its worth from start to finish, equally functional in harming all but some of the fiercest enemies and facilitating access both to key hidden rooms in the overworld and through critical walls in the underworlds. The economy of this item is the best in the game. Unlike the arrow, they don't cost money to use, but Link's limited capacity (eight, to begin with) and the scarcity with which slain enemies offer them force Link to use them somewhat conservatively. The 100-rupee four-bomb capacity upgrades in levels 5 and 7 are both well timed, if a little costly.

Blue Candle - Another great multifunctional item in Link's arsenal. Though with its limited range and once-per-screen use, the candle isn't as useful in combat situations as in its other functions, providing access to secret rooms in the overworld and lighting darkened rooms in the underworld. The only problem here is that Link never has to navigate dark rooms, since they don't appear until level 4 and even the most clueless player has sought out the candle by this point. Perhaps it would have been more interesting to have the layer-cake creatures take Link's fire producing capabilities rather than his magic shield.

Red Candle - This is the most useless weapon in Link's entire arsenal. The candle's upgrade from blue to red simply means you're no longer restricted to one use per screen. It might have been marginally helpful in avoiding some early fumbling around when Link has to burn all those bushes to find secret money giveaways, etc., but by level 7 all you need the candle for is to light up dark rooms, which only requires a single use anyway. You can't even use it to create a protective ring of fire around you because you're not allowed to have more than two lit flames at any given time. Finally, the candle's lighting function and utility against enemies are both rendered obsolete in the very next level when the magic book grants the power of flame, long distance no less, (and also multiple times per screen) to the magic rod.

Letter - Along with the bow and arrow set, the letter is something of a tease, but that's not such a bad thing. More such items would make the game a little too frustrating, but the task of first acquiring the letter makes the potions appropriately more valuable to Link.

Life Potions - Given their high value, the potions should probably be a little more expensive, but their expense, restriction to two uses per refill, the limited number of shops that offer them, (and the prerequisite acquisition of the letter) place them among Link's most cherished items. My only question is why anyone would pay 40 for the one-use bottle when the two-use bottle is only 28 rupees more.

Boomerang - Its functions are three: Kill very weak enemies; momentarily stun other enemies; retrieve rewards (money, energy) from defeated enemies. While the later levels largely prohibit you from implementing the boomerang for either of its first two functions, the last can become crucial when you really need a heart or fairy but can't risk chasing it down in person.

Magic Boomerang - Though it's made available just after the regular boomerang, this weapon's release in level 2 is very well timed. The regular boomerang proves immediately useful, but its limited range quickly becomes frustrating. Endowing Link with a full-range boomerang does not risk making him too powerful too early in the game, as the boomerang only destroys the simplest of enemies, and fails to stun more advanced enemies.

Bow - Finding this in level 1 is a little disappointing if you haven't already stocked your quiver with arrows, but this and the letter are the game's only tease items, and I think their empty utility only makes Link appreciate later items all the more.

Arrow - In contrast to bombs, Link's capacity of arrows is simply the amount of money he has, as they cost one rupee apiece. The arrow is only more effective than Link's sword against a small number of enemies, striking a good balance between cost and effectiveness.

Silver Arrow - This is a pretty lame final weapon. It's a good idea to have a single weapon that defeats the final villain, but as an update to the regular arrow, it rivals the red candle in unimpressiveness, having the same per-arrow cost and being no more effective against non-Ganon enemies than the regular arrow.

Raft - You really want the raft to be cooler than it is. Of all Link's accessories, it's the only vehicle, but once aboard, you don't even get to control it. Water is rather underused in The Legend of Zelda, containing a single enemy and being generally off limits to Link. The raft merely gives you access to level 4 and one extra heart container.

Ladder - Inversely proportional to the raft in attractiveness vs. expected utility, the ladder is at once incredibly useful, especially against non-levitating enemies who can't climb onto the ladder with you. A possible improvement to the ladder's use might have been to contain an overworld item, say, the power bracelet, in an area completely sealed off by thin rivers, so that Link must acquire the ladder before he can venture into this area.

Power Bracelet - A precursor and later complement to the recorder, this facilitates much quicker movement around Hyrule, most notably to one site conveniently located directly below a lake with a life-replenishing fairy. Appropriately, Link has to travel quite far to get it, thereby demonstrating its obvious utility and convenience.

Recorder - Definitely one of the game's cooler items. (So cool, it was borrowed by Super Mario Bros. 3.) The recorder is used right away after its acquisition in level 5 to topple that labyrinth's boss, but once outside level 5, the recorder becomes incredibly useful in allowing Link to travel instantaneously to the entry points of all levels previously completed, as well as to finally reveal the entrance to level 7.

Bait - Another among many reasons (red candle, recycled level boss) that level 7 is just lame. The bait's supposed main function is to attract enemies, making their movements more predictable and thus easier to defeat or avoid. Good in theory, but in reality it's not worth bothering with, and the bait ultimately just amounts to a glorified key when you find you need to sacrifice it to reach the end of level 7, all the more annoying if you haven't purchased it yet.

Magic Rod - This is a great complement to the sword series, possessing attributes both superior and inferior to those of the magic sword. The drawbacks are that it is only as powerful as the white sword, and isn't effective at all against some enemies. But unlike the swords, the magic rod maintains its long distance effects even when you have less than full energy, which is often. The magic rod is also extremely helpful in the moments immediately after Link's sword is paralyzed by those shimmering spheres that show up increasingly often in later levels. The timing of its availability is also well-chosen, as it's the last real weapon Link acquires, and he has three and a half levels in which to brandish it.

Magic Book - While other series weapons (sword, shield, boomerang, candle, arrow) see increased gains in power, range or amount, the magic book updates the magic rod by granting it a completely different function: casting flames wherever it lands. Now the magic rod is even more devastating as a weapon, and you don't have to switch to the candle to light up rooms anymore.

Magic Key - Though the use of keys is a little backwards in Hyrule - "normal" keys can only be used once, on any door, and only "magic" keys can be reused - it's good for Link to first learn to ration his keys and later be freed from this burden. That said, depending on how conservative you are with your keys, the magic key can be one of the least exciting accessories in the game. The careful and thorough Zelda player examines all options before unnecessarily wasting keys, and can build up a surplus of as many as eight or nine by the time this is finally available in level 8.

Blue Ring - Here we see not only Link's weapons evolving, but Link himself. The blue ring is appropriately the game's most costly item, monetarily anyway. Part of the genius of its utility is that it cuts Link's damage in half, while picking up flashing hearts still replenishes his energy by a full container, now equal to four (regular) hits instead of two.

Red Ring - This ring, doubling Link's energy yet again, shows up a little late in the game, and it would be nice to see a red superhuman Link easily slashing his way about the overworld (an event only really necessary if you wish to refill your potion or stock up on rupees for arrow expenditures after unsuccessful attempts at level 9). But the final labyrinth needs some good surprises to offset the difficulty of its vast size, confounding navigational routes and plethora of challenging enemies, and with the silver arrow's previously mentioned disappointing qualities, the red ring fills the role of the final payoff before the game's (anti-?) climactic battle with Ganon.

Thursday, October 25, 2007

Engrish

I saw this on a popular shopping street in Seoul. It's a rather extreme example, but bewildering text like this is everywhere in Japan and Korea. When I came across the awning pictured above, it wasn't actually as shocking as it could have been, because I had already seen this exact storefront on Engrish.com, a wonderful site that catalogues these hilarious phenomena.

How does this happen? Anything in English is COOL. So it's used everywhere in Japan and Korea, even though the vast majority of the target demographic can't understand it.

As mentioned above, http://www.engrish.com has all the goods in this department, but I'll list some more that I've personally encountered.

Yes, the first one is a dentist's office. Tomato Planning provides some kind of real estate service as far as I can tell.

Like with a lot of Engrish, often the interesting part to me isn't so much the grammar errors (tooth/teeth), but the attitude conveyed in the message.

This is a small desk organizer I bought whose sign reads "I will enjoy your space combining various items".

The robot in question was a fifteen-foot figure at a children's museum where many people have their pictures taken.

This cap makes a strong case, but is completely out of touch with the state of Japanese drug use, as far as I know. Penalties for even small user quantities of marijuana or cocaine involve significant jail time, and there isn't any legalization movement to speak of, making it that much stranger that I found this hat in a Sears-like department store.

I learned in Poland that you can't really expect people from totally homogeneous countries to grasp the complexities of American race relations, and I must admit, I'm not sure where I would start if I had to explain to this store's staff why they shouldn't have a rack labeled "black music".