Skip The Water(avc.com)
avc.com
Skip The Water
http://www.avc.com/a_vc/2011/01/skip-the-water.html
16 comments
There's a better way than a star of David, that gets you exactly 4 triangles in 2d using 6 matchsticks.
Do you mean a box with an x through it? I guess the rules never said the ends of the matchsticks had to match up either. I'm definitely enjoying all these different answers which explore (exploit?) the imprecisely specified problem.
That's still more than four triangles -- two right triangles next to each other that way make a larger triangle. One way to make only four triangles is:
\ /
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\1 /
\/
/\
\ / \ /
\ / 2 \ /
---\---/------\----/-
\3/ \4 /
\ \/
/ \ /\
/ \ / \Ok, so make a box with an incomplete x in the middle, a "y", by only sticking that one match partway into the square.
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\----
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|/ |
----No. Sorry to reply so late, I posted so early this morning I forgot about it during the day.
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/ | \
/ / \ \
// \\
/---------\
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[deleted]
the way I read the instructions is that you have to create exactly 4 triangles.
A tangentially related story:
Talented secondary school kids all across America end up doing egg drop competitions all the time. But the rules of each competition are each slightly different. A friend of mine was in a competition which was scored like this:
Score = (intactness of egg, as 1 or 0)/(distance to target)/(mass of egg-protecting device)
While everybody else designed careful egg encasements, but my friend played video games and read sci-fi novels. When it was his turn to test his "device", he grabbed the egg and threw it away from the target, into a nearby bush, which broke the fall and kept the egg intact.
His great distance from the target would have made his score very low. But since he didn't actually have a device, the judges were forced to divide by zero...
Talented secondary school kids all across America end up doing egg drop competitions all the time. But the rules of each competition are each slightly different. A friend of mine was in a competition which was scored like this:
Score = (intactness of egg, as 1 or 0)/(distance to target)/(mass of egg-protecting device)
While everybody else designed careful egg encasements, but my friend played video games and read sci-fi novels. When it was his turn to test his "device", he grabbed the egg and threw it away from the target, into a nearby bush, which broke the fall and kept the egg intact.
His great distance from the target would have made his score very low. But since he didn't actually have a device, the judges were forced to divide by zero...
Awesome, but was he awarded the prize?
I was in a similarly poorly specified contest in the 6th grade. The goal was to take a single large sheet of construction paper and build a paper airplane. The only metric was distance covered and weighting your plane was allowed.
Day of the competition everyone else in class has their planes out, while I'm standing around with a sheet of paper. When it was my turn I promptly balled up the paper as tight as I could around a handfull of 1" steel washers and proceeded to throw the thing across the gym, out the doors and into the hallway outside, roughly 2x the distance covered by the closest competitor.
I was promptly disqualified for unspecified reasons.
I was in a similarly poorly specified contest in the 6th grade. The goal was to take a single large sheet of construction paper and build a paper airplane. The only metric was distance covered and weighting your plane was allowed.
Day of the competition everyone else in class has their planes out, while I'm standing around with a sheet of paper. When it was my turn I promptly balled up the paper as tight as I could around a handfull of 1" steel washers and proceeded to throw the thing across the gym, out the doors and into the hallway outside, roughly 2x the distance covered by the closest competitor.
I was promptly disqualified for unspecified reasons.
I was promptly disqualified for unspecified reasons.
In my experience, most teachers really don't like smart-asses. Devil's advocate will say that the accelerations you submitted your imaginary passengers were way too strong.
My story in that vein was in PE: we were to run 20 minutes at the most constant pace to practice our endurance. The grade was taking into account the regularity of your pace (13 points out of 20 if you hit the exact same time for all but the fastest and slowest laps) as well as your average speed (the remaining 7 points).
Not being the most athletic guys, a friend and I went on our way to run 4 laps of exactly 5 minutes to insure a decent 13 out of 20, where a lap should have been in the 30-45s range. Obviously, our little stunt was not really appreciated. :)
In my experience, most teachers really don't like smart-asses. Devil's advocate will say that the accelerations you submitted your imaginary passengers were way too strong.
My story in that vein was in PE: we were to run 20 minutes at the most constant pace to practice our endurance. The grade was taking into account the regularity of your pace (13 points out of 20 if you hit the exact same time for all but the fastest and slowest laps) as well as your average speed (the remaining 7 points).
Not being the most athletic guys, a friend and I went on our way to run 4 laps of exactly 5 minutes to insure a decent 13 out of 20, where a lap should have been in the 30-45s range. Obviously, our little stunt was not really appreciated. :)
I guess contests like these intend to foster out of the box thinking. But once that very out of the box thinking surfaces every competitor cries foul, including (some) teachers. So, do they perform such competitions because they have to? Or do they just fail to see what they engage them to?
My university had this competition with a different twist:
- The egg was dropped from a 10-story building
- The winner is whoever gets the egg safely to the concrete below in the minimal time. Nothing else matters. (But the judges must be able to see that the egg does indeed touch the concrete).
Most contestants tried to get the egg to drop slowly, e.g. using helium or parachutes.
The winner used a rocket to speed the egg down, and placed the egg within a honey jar. The jar hit the floor and cracked in ~1.5 seconds, the honey absorbed all the shock, and it took 2 more seconds for the egg to roll out of the honey and touch the concrete.
Most contestants tried to get the egg to drop slowly, e.g. using helium or parachutes.
The winner used a rocket to speed the egg down, and placed the egg within a honey jar. The jar hit the floor and cracked in ~1.5 seconds, the honey absorbed all the shock, and it took 2 more seconds for the egg to roll out of the honey and touch the concrete.
I was in a similar egg drop contest in which devices were scored by weight.
The winning entry was helium balloons attached to a basket. It had a decidedly negative weight.
The winning entry was helium balloons attached to a basket. It had a decidedly negative weight.
I love these competitions, and have been captivated by them since I was a kid after seeing a PBS special on MIT's Woody Flowers, who organized them.
And in high school, I designed a project for a state-wide King of the Hill type contest by ignoring the "goal" and focusing on the tie-breaking rules.
The idea was to design a car (using a strict list of household items) to run up a two-sided hill and rotate a long, hanging metal bar against a sensor multiple times. Your opponent also had a bar on the opposite side of the hill. You'd get a point for each time the bar hit the sensor.
Most people spent a significant amount trying to figure out how to rotate the bar. Most points won, after all.
Except that, if nobody scored, the tie breaker was distance; if you made it up the hill further or even made it to the opponent's side of the hill, and nobody "scored", you'd win.
My "car" used the maximum allowed number of mouse traps in an attempt to sling itself over the hill as fast as possible and flip the other car or device over, or at least disrupt the opponent's operation enough so that nobody would score, and I'd win on distance.
Turns out that the scoring objective was so difficult that this was the winning strategy. Still have the car and the t-shirt, but not the $1000 savings bond. And as popular as sports are in the U.S., I did make more competing in nerd competitions than I did on the track team. :-)
And in high school, I designed a project for a state-wide King of the Hill type contest by ignoring the "goal" and focusing on the tie-breaking rules.
The idea was to design a car (using a strict list of household items) to run up a two-sided hill and rotate a long, hanging metal bar against a sensor multiple times. Your opponent also had a bar on the opposite side of the hill. You'd get a point for each time the bar hit the sensor.
Most people spent a significant amount trying to figure out how to rotate the bar. Most points won, after all.
Except that, if nobody scored, the tie breaker was distance; if you made it up the hill further or even made it to the opponent's side of the hill, and nobody "scored", you'd win.
My "car" used the maximum allowed number of mouse traps in an attempt to sling itself over the hill as fast as possible and flip the other car or device over, or at least disrupt the opponent's operation enough so that nobody would score, and I'd win on distance.
Turns out that the scoring objective was so difficult that this was the winning strategy. Still have the car and the t-shirt, but not the $1000 savings bond. And as popular as sports are in the U.S., I did make more competing in nerd competitions than I did on the track team. :-)
I remember back to the days of my youth. My friends and I were participating in Odyssey of the Mind one year, and we had a problem to solve during the state finals.
The goal was to build a bridge that could transport marbles from one side of a 12-inch square to another without anything touching the surface within the square. With a meager set of supplies (postcard, paperclip, rubber band, etc), we scratched our heads for a bit. But in a flash of inspiration, we built a bridge spanning two adjacent sides.
We thought it might've been too clever, so we asked the judges to reiterate the rules. They didn't mention the opposite side. Just another side.
We took home 1st prize.
The goal was to build a bridge that could transport marbles from one side of a 12-inch square to another without anything touching the surface within the square. With a meager set of supplies (postcard, paperclip, rubber band, etc), we scratched our heads for a bit. But in a flash of inspiration, we built a bridge spanning two adjacent sides.
We thought it might've been too clever, so we asked the judges to reiterate the rules. They didn't mention the opposite side. Just another side.
We took home 1st prize.
>build a bridge that could transport marbles from one side of a 12-inch square to another without anything touching the surface within the square
Could you roll the square in to a tube and roll the marbles along what is effectively the underside of the square. It doesn't touch the "[top] surface within the square. Might need 2 paperclips.
The obstacle is the bridge.
Could you roll the square in to a tube and roll the marbles along what is effectively the underside of the square. It doesn't touch the "[top] surface within the square. Might need 2 paperclips.
The obstacle is the bridge.
An interesting thought, but the square was just a masking tape outline that couldn't be messed with.
In a similar class, we were given the problem of designing a large yo-yo. The judgement criterion was the diameter multiplied by the return ratio (how far back up it rebounded). There was no hard-and-fast definition of a yo-yo.
Experimenting with various designs, my buddy and I saw that the biggest challenge was preventing precession, which caused the string to rub against the disks, wasting energy and killing the return ratio.
After plying the prof with leading questions about the design requirements, we submitted a yo-yo made with a single disk, cut from a fridge-sized cardboard box. It had two strings, one running down each side of the disk. That ended precession nicely.
Our entry had both the largest diameter (we had to test it in the stairwell, because the doorframe used to test the other entries wasn't tall enough) and the highest return ratio (in the high-90%).
The best part was that, in response to the predictable storm of objections from the other students, the prof turned it into an object lesson. He made the point that he had never stipulated there had to be one string and two disks, just that there had to be a disk to measure, and a string by which to calculate a return ratio.
Experimenting with various designs, my buddy and I saw that the biggest challenge was preventing precession, which caused the string to rub against the disks, wasting energy and killing the return ratio.
After plying the prof with leading questions about the design requirements, we submitted a yo-yo made with a single disk, cut from a fridge-sized cardboard box. It had two strings, one running down each side of the disk. That ended precession nicely.
Our entry had both the largest diameter (we had to test it in the stairwell, because the doorframe used to test the other entries wasn't tall enough) and the highest return ratio (in the high-90%).
The best part was that, in response to the predictable storm of objections from the other students, the prof turned it into an object lesson. He made the point that he had never stipulated there had to be one string and two disks, just that there had to be a disk to measure, and a string by which to calculate a return ratio.
That's a very interesting solution to the problem. Though as an avid yoyoer, I have to make the point that modern yoyos have solved this problem in an entirely different manner as seen here, in one of the more extreme examples: http://spyy.ca/index.php/pro
I think the hardest part of this advice is training yourself how to see the water.
Reminds me of our bridge making challenge:
Given a set of Popsicle sticks, glue and metal wire we were to build a bridge that spanned 3 feet. The winner was the one that could design the bridge such that the force (uniformly applied in the middle third of the bridge) divided by the squared weight of materials used was the winner.
I had the bright idea to totally minimize wood usage, wrap every single joint in wire then cover it with five layers of glue, and prestress the entire bridge to get the most out of the wire.
Our bridge was so ugly that when we put it down the prof and all the viewers from class just started laughing. We ended up winning (just barely though, someone else had a similar type of hack that I can't remember). Of course we didn't hold that much weight (33 percentile), but ours was by far the lightest.
Given a set of Popsicle sticks, glue and metal wire we were to build a bridge that spanned 3 feet. The winner was the one that could design the bridge such that the force (uniformly applied in the middle third of the bridge) divided by the squared weight of materials used was the winner.
I had the bright idea to totally minimize wood usage, wrap every single joint in wire then cover it with five layers of glue, and prestress the entire bridge to get the most out of the wire.
Our bridge was so ugly that when we put it down the prof and all the viewers from class just started laughing. We ended up winning (just barely though, someone else had a similar type of hack that I can't remember). Of course we didn't hold that much weight (33 percentile), but ours was by far the lightest.
Awesome story, it reminds me of a Dan Pink talk about the candle stick problem.
Dan Pink's TED talk:
http://www.ted.com/talks/dan_pink_on_motivation.html
And a transcript for those who want to skip the video: http://dotsub.com/view/e358ac0c-314e-473a-b051-f0a2deaa3a7b/...
And a transcript for those who want to skip the video: http://dotsub.com/view/e358ac0c-314e-473a-b051-f0a2deaa3a7b/...
I think this advice will be counterproductive for most web entrepreneurs, as I detail here:
http://ourdoings.com/ourdoings-startup/2011-01-13
Summary: In the 2.70 contest, everyone was watching and agreed what the goal was. On the web, the people you want to reach might not be watching, and might rely on authorities who have a different idea of what the goal is. Thinking outside the box may make it harder for people to see that you've solved their problem better.
http://ourdoings.com/ourdoings-startup/2011-01-13
Summary: In the 2.70 contest, everyone was watching and agreed what the goal was. On the web, the people you want to reach might not be watching, and might rely on authorities who have a different idea of what the goal is. Thinking outside the box may make it harder for people to see that you've solved their problem better.
[deleted]
Nothing compares to this scene on "Enigma of Kaspar Hauser": http://www.youtube.com/watch?v=oAnOi0fnxuE
The prof. misremembered the tricky variant of the problem and posed instead a very easy one instead.
The trickier problem is to discern in a single question some arbitrary bit of truth about the world, for example, which of 2 paths leads to the truth teller village (given that exactly 1 does).
No matter the topic (of the bit of info to be discovered), the question necessarily must be answered the same way by the truth and opposite-of-truth teller. This implies that the question has a different meaning depending on whether it's received by a truth or antitruth teller. It does so by making some reference to 'you' (the person asked).*
There's a mechanical solution, no matter what you want to discover. You can build a cumbersome question that spells out an arbitrary truth table explicitly.
For example, "Is it true that ((you're a truth teller and X) or (you're not a truth teller and not X))?" Where you plug in for X: "this path leads to the truth teller village". Turning the expression into unambiguous spoken language is left as homework, but can also be done mechanically.
But this is inelegant and doesn't seem clever. You can pose an equivalent (same output under the stated rules), shorter question, like "If I asked you X, would you answer yes?" The answer will be yes iff X. (because if you ask the liar X directly, he'll say no iff X, but he'll lie about that fact).
You can disguise the technique further by relying on other facts that have a different meaning based on the listener, e.g. referring to "your village" also.
Obviously it's impossible to get 2 bits of information out of one yes/no question - for instance, some arbitrary proposition X, and the polarity of the liar/truth-teller.
* The professor's mistake was to make the goal the very simple "find out if they're a truth-teller or an opposite-of-truth-teller". Most people would arrive first at the simple solution "Do you exist?" (or, equally as informative "are you a tree-frog?" or "is 1+1=2?"). It's easy because all we have to do is ask a question about the world with a known answer, and this lets us know the polarity of the answerer. When your goal is to find out something other than their polarity, the question must be crafted to have the same answer no matter their polarity (be "yes" iff X). When the goal is to know their polarity, the question must have a known answer ("yes" or "no") iff they're a truth teller.
The trickier problem is to discern in a single question some arbitrary bit of truth about the world, for example, which of 2 paths leads to the truth teller village (given that exactly 1 does).
No matter the topic (of the bit of info to be discovered), the question necessarily must be answered the same way by the truth and opposite-of-truth teller. This implies that the question has a different meaning depending on whether it's received by a truth or antitruth teller. It does so by making some reference to 'you' (the person asked).*
There's a mechanical solution, no matter what you want to discover. You can build a cumbersome question that spells out an arbitrary truth table explicitly.
For example, "Is it true that ((you're a truth teller and X) or (you're not a truth teller and not X))?" Where you plug in for X: "this path leads to the truth teller village". Turning the expression into unambiguous spoken language is left as homework, but can also be done mechanically.
But this is inelegant and doesn't seem clever. You can pose an equivalent (same output under the stated rules), shorter question, like "If I asked you X, would you answer yes?" The answer will be yes iff X. (because if you ask the liar X directly, he'll say no iff X, but he'll lie about that fact).
You can disguise the technique further by relying on other facts that have a different meaning based on the listener, e.g. referring to "your village" also.
Obviously it's impossible to get 2 bits of information out of one yes/no question - for instance, some arbitrary proposition X, and the polarity of the liar/truth-teller.
* The professor's mistake was to make the goal the very simple "find out if they're a truth-teller or an opposite-of-truth-teller". Most people would arrive first at the simple solution "Do you exist?" (or, equally as informative "are you a tree-frog?" or "is 1+1=2?"). It's easy because all we have to do is ask a question about the world with a known answer, and this lets us know the polarity of the answerer. When your goal is to find out something other than their polarity, the question must be crafted to have the same answer no matter their polarity (be "yes" iff X). When the goal is to know their polarity, the question must have a known answer ("yes" or "no") iff they're a truth teller.
This similar story is also related to programming. The B-Tree is a "skip the water" idea. Instead of using complex algorithms to balance the tree, it's the tree itself that autobalances, splitting nodes when they're crowded or joining them when they're too empty. That's one of my favourite ideas in programming.
"Skip the water" is a must, especially in sales - it means that you first reach the economic buyers who have the budget and can influence the purchasing decision, and skip / avoid the "See More" type of people who are only obsessive w/ product features!
Actually, I also think this is quite a good story.
It's a nice reminder to check your assumptions.
It's a nice reminder to check your assumptions.
This is a very useful story that everyone should be able to explain.
Many of the worlds' greatest have "skipped the water", so to speak.
-Jesus
-Moses
-Airplanes
-Jesus
-Moses
-Airplanes
There was a Japanese TV program a few years ago testing talented youngsters. One brain teaser was "On this table you have six match sticks of equal length. Arrange them such that you create four triangles."
The match sticks are presented lying flat on a table, and approximately everyone tries to solve the problem in 2D and fails hideously. The right answer, which is a peculiar kind of perverse clever that math teams and startups are fond of, is to make a triangle on the plane and then point the last three match heads upwards from the vertices, creating a pyramid. Each of its four faces is a triangle. Half of the wunderkids got this right.
One kid ran out of time and didn't get it. After being shown the 3D solution, he skipped the next problem entirely and would only say "It works in 2D, it works in 2D, I know it works in 2D" and continue playing with his match sticks.
He eventually solved it halfway through the next problem, confounding the host because his solution was not written in the notes: make two triangles in the plane, flip one, and arrange them to form a Star of David. (Nobody here would call it that, but I'm assuming most HNers know what one looks like.)
Quote the kid: You said I had to make four triangles, but you never said anything about the fifth, sixth, seventh, or eighth.