That was a fun puzzle. I have another one that is math puzzle:
> You are given two eggs, and access to a 100-storey tower. Both eggs are identical. The aim is to find out the highest floor from which an egg will not break when dropped out of a window from that floor. If an egg is dropped and does not break, it is undamaged and can be dropped again. However, once an egg is broken, that’s it for that egg.
> If an egg breaks when dropped from a floor, then it would also have broken from any floor above that. If an egg survives a fall, then it will survive any fall shorter than that.
> The question is: What strategy should you adopt to minimize the number egg drops it takes to find the solution? (And what is the worst case for the number of drops it will take?)
> The aim is to find out the highest floor from which an egg will not break when dropped out of a window from that floor. (emphasis added)
The egg doesn't break when dropped out of the window (i.e., instantaneously). It potentially breaks when it hits the floor after a time delay t. So, the answer is 100 and 0 trials.
I was asked this one in a google phone interview in ~2009. I remember deriving that intervals of 10 floors were optimal in the two-egg 100-story case using some simple calculus and getting that far took me most of the hour. Your solution is super nice, but I hope you don't ask candidates to recreate it :P
That's not optimal though, like the blog says. When doing the first egg, as you get up near the top you've already used up many drops, so the gaps need to get correspondingly smaller.
On August 22, 1994, David Donoghue threw an egg out of a helicopter onto a golf course in the UK, from a height of 213 meters (700 feet). He now has the record for the longest egg drop without breaking in the world (all without an outside structure for added protection!). https://www.scienceworld.ca/resource/egg-drop/
Eggs are evolutionarily designed to survive falling out of nests and perches. You assume that the egg is dropping onto a hard surface, but there is no mention of that in the puzzle.
That's even assuming it's a chicken egg, but other eggs may be more resilient. It may not even be a biological egg, but an artifically designed egg, that is resilient to drops onto harder surfaces.
I bet a human egg would survive undamaged even when dropped from 30,000 feet.
(Following Haldane's observation that "You can drop a mouse down a thousand-yard mine shaft and, on arriving at the bottom, it gets a slight shock and walks away. A rat is killed, a man is broken, a horse splashes.")
“Hypotheses proposed for the adaptive function of egg shape typically invoke a decrease in egg loss for cliff-nesting birds laying conical eggs that roll in a tight circle; an increase in incubation efficiency when egg shape is associated with the number of eggs in a clutch; or other advantages related to strength, diet, and development. For example, spherical eggs might be advantageous because the sphere is uniformly strong and would be robust to incidental damage in the nest.
Spherical eggs, with their minimal surface-area-to-volume ratio, also require the least amount of shell material for a given volume and possibly optimize gas exchange by providing a large surface area for pores. In contrast, conical eggs may be beneficial because they can accommodate an increased concentration of pores at the blunt end, creating a specialized respiratory site for accelerated neural development in precocial birds. Moreover, conical eggs may protect the blunt end (from which chicks usually hatch) from debris contamination or, in colonial breeders, increase resistance to impacts because a larger proportion of the eggshell is in contact with the substrate. Finally, it has also been proposed that adaptations for flight influence egg shape indirectly through the morphology of the pelvis, abdomen, or oviduct.”
and concludes
“Our macroevolutionary analyses suggest that birds adapted for high-powered flight may maximize egg size by increasing egg asymmetry and/or ellipticity, while maintaining a streamlined body plan”
It doesn’t even mention “better protection against drops”.
plus, there's a cost to the parent in making the egg shell arbitrarily drop-proof, so there's a balance of resources required to survive drop versus genetic benefit of surviving the drop
That’s exactly what I said. “Selects out” is the terminology used when discussing evolution. It’s not an active “decision,” it’s a thing that happens. “Natural selection.”
Ok. But what does that mean in the context of the comment that you replied to? (Which doesn’t talk about decisions or nature chasing things, at least as it stands now.)
Is a harder shell which has higher costs than benefits selected out? Maybe we’re all repeating the same thing.
Ah I get what you’re saying. My point is that yes, some other type of shell probably selected out and this one wasn’t. Maybe it was dumb luck, maybe it was a slight advantage in a very specific condition that allowed it to persist, but ultimately we can only theorize.
I would understand the selective advantage if it made the egg shell more resistant to accidental breaking within the nest. But for birds that nest in tree branches or perches, what advantage is there for an egg resisting a fall without breaking? That egg is lost anyway; for most bird and egg shapes, there's no way to bring it back to the nest.
So this is a pretty common mistake people make when discussing evolution. It’s not about what is advantageous, it’s about what survives the selection process. A combination of odds that play out.
Our eyes are not advantageous or chosen by evolution, our eyes were just acceptable enough for us to survive while other variations were not. There was no decision or work/progress towards an end goal. A creature developed along a certain line by chance, the situation in nature was such that they survived and passed on that trait, while others did not. Natural selection in action.
No, I understand how natural selection works. I'm a fan of Stephen Jay Gould.
Please keep in mind the original comment I was responding to:
> Eggs are evolutionarily designed to survive falling out of nests and perches.
My response is "no, they are not". First, like you said, they are not "designed" in any way. But that's the pedantic response. Second, there's no selective pressure for eggs resistant to dropping from a nest, since they are lost anyway and won't pass any genes. It's likely that something else is being selected for -- or at least, that the more resistant shell is not harmful to reproduction of the species -- but there's no selection for nest-drop-resistant eggs because that confers no advantage. Maybe it's just harmless, which can make the trait survive. See the difference?
> There was no decision or work/progress towards an end goal.
It helps to assume everyone understands this, otherwise we cannot get to the meat of the discussion without nitpicking every single sentence.
"Instead of giving students realistic situations that they could analyze, textbook authors began to fill books with make-believe contexts – contexts that students were meant to believe but for which they should not use any of their real-world knowledge."[0]
I think for a math puzzle that people choose to do for fun, pseudocontext is fine. We can suspend our disbelief and engage with the math as it's intended to be engaged with.
In a classroom setting, on the other hand, I think it's best to stay away from pseudocontext. Either make the context realistic, make it clearly fantastical, or just use math as the context without trying to dress it up in real-life details.
Pseudocontext in a teaching environment is problematic because it requires learners to ignore the common-sense reasoning that they'd apply to a real-life problem and it reinforces the impression that math has nothing to do with the real world.
In this case, we could call it a dragon egg and that would be enough I think to transform it into a fantasy context, since nobody has preexisting assumptions about how durable a dragon's egg might be.
For one, I think the egg experiment is already sufficiently fantastical and its rules follow logically.
Ignoring common sense reasoning is an important skill. We need rigorous answers where they are available, and the common sense answer is almost always very wrong.
I still agree that school textbooks take this much too far in many cases, but this riddle is fine.
Except that I think it penalizes the neuro-diverse mind that can't get over the fact that this puzzle doesn't make rational sense, and they get stuck on that, at the expense of doing the problem.
Eggs can absolutely survive very far falls. You are making a lot of assumptions here. Like:
- This is a typical bird's egg. Most insect eggs for instance can survive any fall.
- This is a hard floor, not water, snow,...
- There is nothing to protect the egg.
- Floors are typical ~3m floors, not floors from a doll house.
- The experiment is done on earth
The point of maths is that it doesn't need to be grounded into the world. It is a "if A then B" machine. If A applies to a world then B will apply to that world too, and it will always be the case, it doesn't depend on experimental results.
It is great because scientists, engineers, etc... only have to show (with their experiments and measurements) that A applies to our world, and maths automatically tell them that B applies too, no need for extra experiments and measurements, except for verification.
So in this problem, if you use your "real world" experience with eggs, you only solve the problem for a limited set of preconditions. If you solve it properly, using maths, then it will work in any situation that matches the data of the problem, even for people of Pluto.
Also, if you want to talk "real life", I am not sure a 100 story building exists (there is 101, but exactly 100?). And if that building exists, does it have windows you can drop an egg from at every floor?
I guess the issue is that you assumed the type of egg and jumped to a conclusion about the egg you assumed was used in the experiment. You also assumed the environment - who is to say that the environment is earth? Etc etc.
The point of the thought experiment is to get the non-math-inclined thinking about how they would try to get to the desired result within a limited set of tries: Practical considerations are to be ignored specifically for this purpose.
By going against the intentions of the thought experiment and blurting out "0 floors, eggs can't survive very far falls", the individual is deliberately trying to stamp out creativity & problem solving skills, as well as any attempts at getting others thinking about their approaches to the aforementioned problem. To them, the strict adherence to practical reality must be imposed 100% of the time, and there should be no time for playful thinking.
This was also my first thought, as an admitted deficient with any math beyond (and sometimes including) arithmetic. My second thought was: did you not consider that eggs differ from one another?
I hate how you are downvoted for an accurate statement. I love math, but I've come to understand from a teacher's perspective how accurate your statement can be. I liked your insight.
I’m not one of the downvoters but I suspect he’s being downvoted for an inability (or refusal) to think in hypotheticals for the sake of an interesting problem. It gives the impression of someone who so dislikes thought exercises or is too insecure about their ability to solve them that they prefer to ruin them with silly objections about practicality.
No, what I think he is highlighting is how people not interested in math approach a problem like this. I asked my girl friend this question on the way home from our Mother's Day dinner, and she had a very similar statement. It says something about how mathematicians could improve on their analogies, not whether he is interested in solving a math problem. I found it insightful even if it was not directly tied to the "correct" speech on Hacker News.
That's not what this is about though. It's about sparking interest in math, and many on here are showing exactly why math people can me off putting for non math people.
It doesn’t really have anything to do with math. It’s problem solving. I’m pretty sure the legendary farmer who had to get his fox and chicken across the river never actually existed, but the point of the problem is to be a framework for using your brain, not an actual attempt to help a farmer with a real dilemma.
Maybe someone doesn’t enjoy these types of problems, but objecting based on the obviously-contrived premise not being realistic just makes them look dumb.
Did I miss it, or does the article not actually explain why we know that at least one pair can be eliminated every round? It seems plausible that the process could get "stuck" in a place where Peter and Sandy both don't learn anything new from discovering the other doesn't know the answer yet.
(The puzzle being solvable means that they can't actually get stuck, but that feels like outside information...)
You’re totally right: there’s no guarantee that the candidate list continues to shrink, and there’s no guarantee that it won’t get stuck. For this problem, we are able to leverage the fact that it ends after 15 rounds, so the problem is less “run it until it finds a solution” and more “find the solution after exactly 15 rounds”.
In fact, if you let it run past 15 rounds (for N=100, at least), and check the number of disqualified candidates between rounds, you’ll find that it does get stuck in after 15 rounds as the candidate list doesn’t change. The flip side of this is that there are also solutions before 15 rounds, too, but with the problem’s original statement, we’re looking at exactly 15.
Yup, without a critical assumption - each "I don't know eliminates a potential solution" - it's not solvable because there is no new information when someone says "I don't know the answer"
The only critical assumptions are that they are aware of the details of what both of them have been given and that if they could have known the answer they would say so (i.e. that they are being perfectly logical and not making a mistake). The fact that this eliminates potential solutions is a consequence of that and the problem as set out. The whole puzzle hinges the fact that one person not knowing the answer from the part they are given (and the knowledge that the other person does not know after each step) constrains (i.e. gives information on) the answer. (but it is correct that as stated you could not derive all pairs of numbers this way: the puzzle gives you a sequence for which the pair is knowable)
For N=100, though, they could know beforehand that potential solutions are eliminated at least the first 15 times someone says "I don't know". For any N, it should be possible to work out the longest such a game could go on (until one of them says, "I don't know, and I don't know any more from your last statement").
I would assume it hinges on the fact that when one party says "I don't know the answer" then they are eliminating a possibility (which isn't intuitive at all with that answer).
I believe (I could be wrong here, I haven't rigorously checked this) that when the numbers are less than 100 and both parties know this, there will always be a possibility eliminated by a "I don't know" statement.
After the round in the posed problem, it reaches a stalemate.
If Peter had said "I don't know the numbers" at the end there, Sandy can exclude the possibility that the numbers were 77 and 84. However, at this point Peter already knows that Sandy still doesn't know the numbers: Even if the sum Sandy was told was 161, there are still five remaining live possibilities for that sum (62 + 99, 65 + 96, 70 + 91, 76 + 85 and 80 + 81), so Peter gains no new information from Sandy's next statement that Sandy does not know the numbers.
> A census taker approaches a woman leaning on her gate and asks about her children. She says, "I have three children and the product of their ages is seventy–two. The sum of their ages is the number on this gate." The census taker does some calculation and claims not to have enough information. The woman enters her house, but before slamming the door tells the census taker, "I have to see to my eldest child who is in bed with measles." The census taker departs, satisfied.
There’s actually many variation of those Peter and Simon/Sally problems. I’ve seen with two, three and up to four numbers. Often, one of them claim they (finally) know the answer after a certain number of interactions and _then_ the other claiming that they either can, or can’t figure it out. “I knew that” or “Oh, really?” respond the one who knows, and finally: “Then I know what those are.”
One can include certain clues, like “Those are three beautiful numbers” to indicate those are different, or “You would like them” if either of them is presented as a fan of prime numbers, perfect numbers or something like that. With children’s age, they might be in school, boarding school, went to university, or have children, to indicate they are below or above 6, 14, 18, etc.
Hmm. I don't like how "I have an eldest child" is used to imply "therefore my second-oldest child can't have the same age [when rounded down to a year number]". As the article says: "although one could be a few minutes or around 9 to 12 months older". I could imagine parents objecting to the claim that one twin is older than the other (or even saying they didn't keep track of which was born first); but a child having been born 10 months before another is something I'd expect parents to remember, and I don't think it would be weird in that case to call the older one the "eldest".
In case anyone else is confused like I was, the article is wrong about Sandy's first round. The article claims she can eliminate (1, 4) on the basis that it's the only pair that adds up to 5, but the author forgot about (2, 3), which was not eliminated by Peter's first round because it has the same product as (1, 6).
The algorithm itself is correct, and the answer it arrives at is correct.
Oh good catch, I even crossed out the right ones in the above code block. Edited to be (2, 2), which is a better example (since (1, 3) was already cleared out). Thanks!
Two people are told rational numbers of a particular form. They want to know whose number is larger:
> Albert I don’t know.
> Bernard Neither do I.
> Albert Indeed, I still do not know.
> Bernard And still neither do I.
> Cheryl Well, it is no use to continue that way! I can tell you that no matter how long you continue that back-and-forth, you shall not come to know who has the larger number.
> Albert What interesting new information! But alas, I still do not know whose number is larger.
They keep saying “No matter how much we do this, we still won’t know” and eventually they know. It’s great and I recommend reading it.
I prefer this less sterile framing of it. It was the most fun that I ever had with a puzzle, so to anyone scrolling around on this page, I would recommend not jumping straight to the solution :)
After the elimination of candidates due to Peter's first comment, the entry in the product map with value 7920 should also have the candidate (80,99). The whole point of this step is that there should be no single-candidate entries left in product.
This isn't quite right. This does find the solution when the problem is well-formed, but it doesn't prove that the solution is valid:
if round == 15:
for product in products:
if len(products[product]) == 1:
print('Peter: I do know the numbers')
print(products[product][0])
return
This should not return - it should continue iterating. If the problem (and the code) is correct, then it should only find one answer - but unless you exhaustively search and find exactly one answer you don't know that it's correct.
Incidentally, I was also a bit surprised by
candidate_pairs = set()
for i in range(1, N):
for j in range(1, N):
if (i, j) not in candidate_pairs and (j, i) not in candidate_pairs:
candidate_pairs.add((i, j))
That first (i, j) check can never return false and should be omitted, right?
The ‘normal’ way to write something like that (I hope I got the range ends right) is
for i in range(1, N):
for j in range(1, j + 1):
candidate_pairs.add((i, j))
Also, conceptually, you don’t have tuples, but multisets. If you were to use those, you wouldn’t need that if at all.
Multiset isn’t built-in to python, so you’d need to use an external package. Alternatively, write a class for storing pairs of ints i, j that enforces i ≤ j.
That’s not a multiset, it’s a set containing tuples. A multiset (https://en.wikipedia.org/wiki/Multiset) allows you to add identical items multiple times.
You would use that to, for example, store the knowledge that 6 can be written as 3+3, by using a multiset {3,3} instead of a tuple (3,3)
That's a good point! The `15` check relies on 1) knowing it will stop after 15 rounds as written in the problem (though I explore that later) and 2) assuming that there must be a single answer after 15 rounds (which I also explore later).
And that's true, the `(i, j)` check is spurious - I blame it on it being throwaway blog post code :)
Regarding the second snippet, yeah, that (i,j) test will always be true. And if the inner loop is changed from `range(1, N)` to `range(i, N`) then it will be totally unnecessary anyways to have the conditional. With that change every pair would be generated exactly once.
This puzzle also exists in a transfinite version, where the two players converge on the correct solution but require longer than infinity for the process:
I understand it’s a puzzle and it requires some suspension of disbelief, but I think Peter is wrong. He doesn’t know the numbers after 14 tries. He is assuming that Sandy has figured out the trick and is able to correctly keep track of 9801 + 198 = 9999 lists of number pairs in her head. As far as I can tell by how the puzzle is phrased, that’s a totally unreasonable assumption. It’s more likely that she says “I don’t know” because she has no idea how to approach the problem, and the fact that that is even possible means that Peter is not really getting any information from her at all.
It’s generally presented as both of them being experienced mathematicians who have been doing it for a while, that they have access to a notebook and a calculator and recently (I’ve seen those since the 1980s) to a computer.
We are given that he says he does know and have no reason to doubt it. It is possible that Peter and Sandy know one another and already know that they are perfect at figuring out maths/logic puzzles.
The puzzle doesn't state whether Sandy has figured it out because that information is not necessary to figure out the solution. We can infer that she has (under that implicit assumption that she is a perfect logician) by the fact that we know less than she does and still have enough information, though. As for that last part: we also don't truly know that Peter isn't lying just to mess with the puzzle, I guess. That is another implicit assumption.
Bonus meta-puzzle: the puzzle as stated is actually unsolvable. Both Sandy and Peter have to know something that the puzzle implies but does not actually stipulate that they know. What is it?
That the other person is following the same system as them to decide whether they "know the numbers". It would be super easy for one of them to say they don't know for some other reason, in which the others' reasoning would be false.
This is what I thought. Two unstated (but implied) parts of the puzzle are that they can’t tell each other the numbers outright, and that they can only share information in yes/no format. But since they have clearly already agreed on a system beforehand, they could have also encoded the digits in binary or something.
So each person could have been sharing one binary digit of their number with each round. Or they might have used a completely different encoding. We simply cannot know.
process of elimination. as soon as peter starts, all primes are off the table. Given that, some number pairs which sum to the same number as just one other number pair where the second number pair has at least one prime are off the table. And so on and so forth. My instinct was to program the solution. Bet there are people who could do it in their heads.
It's only the primes above 50 that are immediately eliminated by the fact that Peter does not know the answer right away.
The primes below that can be present in products that have multiple possible answers: eg. we can't eliminate 47 right off the bat because Peter could have been told a product like 3290.
>as soon as peter starts, all primes are off the table
How do you figure? If Peter is given 97 (a prime), how is he supposed to know that a) he's been given a product (puzzle doesn't say he was told it's a product) and b) if he knows he's been given a product how is he supposed to know 97 is prime? He wasn't told that in the puzzle either.
That the order of the numbers is unimportant? Otherwise every non-duplicate solution has a correlate solution, inverted. E.g. (3,2) implies (2,3) is also a solution.
Common knowledge is a special kind of knowledge for a group of agents. There is common knowledge of p in a group of agents G when all the agents in G know p, they all know that they know p, they all know that they all know that they know p, and so on ad infinitum. It can be denoted as {\displaystyle C_{G}p}.
-- Create Table #x(v) from 1..99
WITH x AS (SELECT n FROM (VALUES (0),(1),(2),(3),(4),(5),(6),(7),(8),(9)) v(n))
SELECT ROW_NUMBER() OVER (ORDER BY (SELECT NULL)) AS v INTO #x
FROM x ones, x tens ORDER BY 1
;
DELETE FROM #x WHERE v > 99
;
-- Create Candidate #c(x, y, s, p) with pair (x,y) (x <= y) and sum `s` and product `p`
SELECT x.v AS x, y.v AS y, x.v + y.v AS s, x.v \* y.v AS p INTO #c FROM #x x, #x y WHERE x.v <= y.v
;
-- Peter: I don’t know the numbers.
DELETE FROM #c WHERE p IN (SELECT p FROM (SELECT p, COUNT(*) AS n FROM #c GROUP BY p) AS d WHERE n < 2)
-- Sandy: I don’t know the numbers.
DELETE FROM #c WHERE s IN (SELECT s FROM (SELECT s, COUNT(*) AS n FROM #c GROUP BY s) AS d WHERE n < 2)
-- Peter: I don’t know the numbers.
DELETE FROM #c WHERE p IN (SELECT p FROM (SELECT p, COUNT(*) AS n FROM #c GROUP BY p) AS d WHERE n < 2)
-- Sandy: I don’t know the numbers.
DELETE FROM #c WHERE s IN (SELECT s FROM (SELECT s, COUNT(*) AS n FROM #c GROUP BY s) AS d WHERE n < 2)
-- Peter: I don’t know the numbers.
DELETE FROM #c WHERE p IN (SELECT p FROM (SELECT p, COUNT(*) AS n FROM #c GROUP BY p) AS d WHERE n < 2)
-- Sandy: I don’t know the numbers.
DELETE FROM #c WHERE s IN (SELECT s FROM (SELECT s, COUNT(*) AS n FROM #c GROUP BY s) AS d WHERE n < 2)
-- Peter: I don’t know the numbers.
DELETE FROM #c WHERE p IN (SELECT p FROM (SELECT p, COUNT(*) AS n FROM #c GROUP BY p) AS d WHERE n < 2)
-- Sandy: I don’t know the numbers.
DELETE FROM #c WHERE s IN (SELECT s FROM (SELECT s, COUNT(*) AS n FROM #c GROUP BY s) AS d WHERE n < 2)
-- Peter: I don’t know the numbers.
DELETE FROM #c WHERE p IN (SELECT p FROM (SELECT p, COUNT(*) AS n FROM #c GROUP BY p) AS d WHERE n < 2)
-- Sandy: I don’t know the numbers.
DELETE FROM #c WHERE s IN (SELECT s FROM (SELECT s, COUNT(*) AS n FROM #c GROUP BY s) AS d WHERE n < 2)
-- Peter: I don’t know the numbers.
DELETE FROM #c WHERE p IN (SELECT p FROM (SELECT p, COUNT(*) AS n FROM #c GROUP BY p) AS d WHERE n < 2)
-- Sandy: I don’t know the numbers.
DELETE FROM #c WHERE s IN (SELECT s FROM (SELECT s, COUNT(*) AS n FROM #c GROUP BY s) AS d WHERE n < 2)
-- Peter: I don’t know the numbers.
DELETE FROM #c WHERE p IN (SELECT p FROM (SELECT p, COUNT(*) AS n FROM #c GROUP BY p) AS d WHERE n < 2)
-- Sandy: I don’t know the numbers.
DELETE FROM #c WHERE s IN (SELECT s FROM (SELECT s, COUNT(*) AS n FROM #c GROUP BY s) AS d WHERE n < 2)
-- Peter: I do know the numbers.
SELECT * FROM #c WHERE p IN (SELECT p FROM (SELECT p, COUNT(*) AS n FROM #c GROUP BY p) AS d WHERE n < 2)
/*
x y s p
77 84 161 6468
*/
But there's not a writeup there. That's just a statement of the problem.
This is not the hardest logic puzzle in the world, though stating the answer can be tricky. If it takes one day for a solitary blue-eyed person to realize he's the one with blue eyes and leave, then it takes 100 days for a group of 100 blue-eyed people to realize they have blue eyes and leave.
The blue eyes puzzle relies on everyone receiving input from the same synchronized digital clock ("Every night at midnight, a ferry stops at the island"), which is unusual for a logic puzzle. The puzzle here is similarly discretized, but it's more a pure question of sequence - each line of dialog happens after the previous line, and that's all that matters.
import itertools
import collections
candidates = [s for s in itertools.product(range(100), range(100)) if s[0]<=s[1]]
def solutions_filter(sols, agg, flip=False):
aggregates = [agg(s) for s in sols]
d = collections.defaultdict(lambda:0)
for s in aggregates:
d[s] = d[s]+1
if not flip:
return [s for s in sols if d[agg(s)] != 1]
else:
return [s for s in sols if d[agg(s)] == 1]
ns = candidates
for i in range(7):
ns = solutions_filter(ns, lambda x:x[0]*x[1])
ns = solutions_filter(ns, lambda x:x[0]+x[1])
print(solutions_filter(ns, lambda x:x[0]*x[1], flip=True))
(Usually, next step is variable names, comments, etc.)
It's worth looking at the other solutions in-thread. Key points:
- Use of higher-order operations, like filters and aggregations, to avoid loops.
- Passing around functions (the * and + side are the same, except for one operation) to avoid repeated code
This results in:
- Less code. Defects per KLOC tends to be pretty constant, so shorter / higher-level programs are typically less buggy.
- In particular, less repeated code. Cut-and-paste introduces a whole slew of potential defects.
- Less dependence on order-of-operations, which eliminates whole classes of errors, from off-by-ones to data structures in intermediate inconsistent states.
> You are given two eggs, and access to a 100-storey tower. Both eggs are identical. The aim is to find out the highest floor from which an egg will not break when dropped out of a window from that floor. If an egg is dropped and does not break, it is undamaged and can be dropped again. However, once an egg is broken, that’s it for that egg.
> If an egg breaks when dropped from a floor, then it would also have broken from any floor above that. If an egg survives a fall, then it will survive any fall shorter than that.
> The question is: What strategy should you adopt to minimize the number egg drops it takes to find the solution? (And what is the worst case for the number of drops it will take?)
I wrote up a solution for this (along with a generalized analytical solution) on my blog: https://blog.kdheepak.com/the-egg-tower-puzzle