They're interesting to adults, too! Simple enough that it feels like you should be able to blurt out the answer, I'm more than twice the maximum recommended age and a professional engineer, but (at least for me) it takes some thought. The top recommended three:
> 1. Masha was seven kopecks short to buy a first reading book, and Mishalacked one kopeck. They combined their money to buy one book to share, but even then they did not have enough. How much did the book cost?
> 3. A brick weighs one pound and half the brick. How many pounds does the brick weigh?
> 13. Two volumes of Pushkin, the first and the second, are side-by-side on a bookshelf. The pages of each volume are 2 cm thick, and the cover – front and back each – is 2 mm. A bookworm has gnawed through (perpendicular to the pages) from the first page of volume 1 to the last page of volume 2. How long is the bookworm’s track?
I do take objection to the answer to question 13 - the author seems particularly set on one way of loading the bookshelves as correct.
> A brick weighs one pound and half the brick. How many pounds does the brick weigh?
I am a native english speaker and am having a hard time parsing this one. The only sane interpretation I can think of is that one pound + half the brick = the whole brick.
EDIT: I think the reason it is so confusing to me is because "and half the brick" sounds like (the start of) an independent thought. "A brick weighs one pound and half the brick was painted yellow".
This version is much clearer, IMO: "A brick weighs one pound plus half a brick". Maybe there is a fear that wording the problems too clearly makes the solution obvious.
These problems are worded to be deliberately confusing, especially #1. Is it a translation issue or are they worded because the math itself is too obvious once the wording has been deciphered?
Here’s a way to use algebra to grind out the solution to #1 with no particular insight needed.
Assume that prices and the amount of kopeks a person has are both represented by nonnegative integers. Let A be Masha’s kopeks, let I be Misha’s kopeks, and let B be the price of the book. We are given the following:
A = B - 7 (1)
I = B - 1 (2)
A + I < B (3)
Substituting (1) and (2) into (3) yields
B - 7 + B - 1 < B (4)
This simplifies to
B < 8 (5)
B = 7 satisfies (5) and, from (1) and (2), implies that A = 0 and I = 6, which together satisfy the givens (1), (2), and (3). So B = 7 is a solution. Furthermore, we cannot have B < 7 or else (1) would imply A < 0, contradicting the assumption that our variables are represented by nonnegative integers. So B = 7 is the only solution.
Couldn't the book cost 7.5k and one has 6.5 and the other has 0.5? Along those lines, isn't anything in the range of costing 7->8 (non-inclusive) acceptable (e.g. 0.9k and 6.9k)?
That’s a nice approach. (It’s the same one given by bencollier earlier: https://news.ycombinator.com/item?id=27885681). I regard it as requiring a bit of insight, as opposed to my approach, which is more like grinding gears to reach a conclusion.
This is one of my main observations growing up with math: it's the moments of beauty and elegance that are the most exciting, but the grinding gears thing is also a necessity. They complement each other. For instance when you're just learning the basics there's a lot of these "wow what an insight" but over time you figure out that people have distilled it into a mechanical procedure, which also has some attraction to it. Something like quadratic equation turns the search for a pair of numbers that add up some one thing and multiply to another into a simple formula. You then use that mechanism to build ever more elaborate ones.
I came to the same conclusion the same way but it felt wrong due to the phrase "They combined their money to buy one book to share". Perhaps the phrase lost something in translation.
Yes this is the flaw in the question. They should have said that the sum of their money is insufficient. Combine implies a physical action which can't happen if one of the parties has nothing.
I suppose the question doesn’t mention that volume 1 is on the left and volume 2 is on the right but I guess that would be assumed by any speakers of left to right languages.
> I suppose the question doesn’t mention that volume 1 is on the left and volume 2 is on the right but I guess that would be assumed by any speakers of left to right languages.
The answer is supposed to be 4mm; the only way for that to work if volume 1 is on the left is for the bookworm to gnaw its way out of the book from v.1.p.1, cross the outside of the two books without gnawing anything until it reaches the back cover of volume 2, and then gnaw its way through that cover to reach the final page of volume 2.
I don't think that's what the question has in mind. The point of being a bookworm is that you don't leave the book. So the answer would appear to require that volume 2 is shelved in front of volume 1. I don't know why that would be the case.
(0-indexing of the pages for fun, plus it fit better, also reminded me of annoying protocol specs that mix 0- and 1-based indices with different elements)
The first page of V1 is the rightmost page (shelved) of Volume 1, and the last page of V2 is the leftmost page (shelved) of Volume 2. So the bookworm ends up going only through the covers. Having volumes shelved in order from left-to-right is conventional in left-to-right languages since that's the same direction we read, and you'd want to "read" through the titles to find the volume you wanted.
Wow, this relies on both books being in the same orientation, with front cover to the right. It assumes a lot. For perspective, I for years kept books shelved upside down because that orientation was easier for me when reading spines.
I guess it relies on the books being ordered and arranged the same way they'd be in every single bookstore and library in the world (in left-to-right ordering countries).
But it's true, maybe this is too much to assume. Most of the time when I've seen this puzzle it's shown the book spines in an image to make it clear, and many people still can't get it. Then again, that would rely on knowing whether it was using top-to-bottom or bottom-to-top book title orientation, so perhaps the only solution is for the author to spell out "the first page of volume 1 is next to the last page of volume 2."
> in every single bookstore and library in the world
Not so fast. Some cultures (Japan for one, I think China and Taiwan as well?) have page-ordering right-to-left but books are generally stacked left-to-right from what I've seen (and in ether case bookstores don't order volumes differently depending on if it's native right-to-left books or foreign right-to-left ones).
German books have the orientation of the writing on the spine flipped. I don't like storing books upside down, so it makes a mess in my mixed English and German bookshelf.
The point with this question is that if volume 1 is on the left and volume 2 is on the right, the first page of volume 1 is facing right and the last page of volume 2 is facing left, so the only thing between them is the two covers. Hence, the answer is 4 mm.
The problem statement gives us that there are 2cm of pages in each book. So they are not empty. The confusion is in which order the books would be on the shelf, and consequently which direction the bookworm would be moving and through what.
The bookworm could have moved right to left from volume 1 to volume 2.
Assuming the spines are facing out, the books are right-way up, and volume 2 is on the left of volume 1, then the answer would be 44mm.
While the answer does assume that V1 is on the left, there's no contradiction in your statement. If V2 happened to be on the left, it would still be perfectly logical for "a worm eat perpendicular to the pages and go from the first page of volume 1 to the last page of volume 2." They would simply have to go through more pages.
My answer to #1 is less than 8 kopecks, and Masha has less than one. There's a problem: nowadays kopecks are the minimal unit of currency. Either it means you have to think of the old Imperial money units (polushka, 1/4 of kopeck), or think of fractional amounts of money.
I assumed kopecks were pennies. If Misha needs one, and Masha doesn't have enough to give her one, than Masha must have none. So the answer follows from that.
1. Depends on whether kopecks are divisible into a smaller monetary unit or not. If they are divisible into 100 units, I believe the answer is "anywhere between 7.00 and 7.99 kopecks".
I think part of the point of this brochure is to think about the problems intuitively in the context they are presented. So in the first problem it's just kids trying to buy their first book, it would be silly to think Masha had a fraction of a kopeck (assuming you understand what a kopeck is, I really think it should have been translated as cent) and that the answer could be in range [7, 8). This may be what he talks about when he says that many academics fail at these problems.
Similarly, in problem #2 the cork indeed costs 0.5 kopecks but in this case we're just thinking about cost conceptually, not in terms of how much money a person actually has on hand.
Indeed, but it likewise seems intuitively reasonable to think that a book costs much more than 7 cents (or 7 times whatever the atomic unit of currency is) and that over 100 times the atomic unit is more reasonable.
Works even if Kopecks are divisible: say the books value is 7.5, Mash must have 0.5 (7.5 - 7) and Misha 6.5 (7.5 - 1), however now when you combine them they sum to exactly 7.5 not less than, the only way to arrive at less than is if Masha has 0. So its always exactly 7
> say the books value is 7.5, Mash must have 1.5 (7.5 - 7) and Misha 6.5 (7.5 - 1), however now when you combine them they sum to exactly 7.5 not less than
There are several problems with this:
- 7.5 - 7 is 0.5, not 1.5
- 1.5 + 6.5 is 8, not 7.5
- 0.5 + 6.5 is still less than 7.5
The problem specifies that 2x - 8 < x. There is no way to constrain this to the specific solution x = 7. Everything would work fine if the book cost -2.6 kopecks.
> 1. Masha was seven kopecks short to buy a first reading book, and Mishalacked one kopeck. They combined their money to buy one book to share, but even then they did not have enough. How much did the book cost?
> 3. A brick weighs one pound and half the brick. How many pounds does the brick weigh?
> 13. Two volumes of Pushkin, the first and the second, are side-by-side on a bookshelf. The pages of each volume are 2 cm thick, and the cover – front and back each – is 2 mm. A bookworm has gnawed through (perpendicular to the pages) from the first page of volume 1 to the last page of volume 2. How long is the bookworm’s track?
I do take objection to the answer to question 13 - the author seems particularly set on one way of loading the bookshelves as correct.