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>I think it is ok as a brain teaser, but there will probably be one kid in the class to see it and all the other kids feel dumb or whatever.

If you see the trick, you save some time on the calculation. If you don't you have to add some more numbers. It's not that difficult.

Yes, but I am worried if a school uses question which can be solved with a trick that the whole instruction in class starts to revolve around tricks (which naturally will happen since the one kid who sees it will answer first).

This trick discussion reminds me of the great anectode about von Neumann:https://news.ycombinator.com/item?id=5950755

Tricks? It's not tricks it's word problems, which is just as important as any other problem. You need to relate math to the real world and having a equation doesn't help, having stuff like Susan is taking a train at 2:32 for 50 kms going 60km/h what time did she get off helps people understand how to turn a real world problem into a math equation to solve it.

Without the trick you get:

  Coffee concentration before first,second,third,fourth drinking = 1,x,y,z
  x = 1/2
  y = x*2/3 + 0*1/3 = 1/3
  z = y*5/6 + 0*1/6 = 5/18
  
  Coffee drunk = 1/2 + 1/3 x + 1/6 y + z
    = 1/2 + 1/6 + 1/18 + 5/18
    = 3/6 + 1/6 +     2/6
    = 1
  Milk drunk = 0 + 1/3(1-x) + 1/6(1-y) + (1-z)
    = 1/6 + 2/18 + 13/18
    = 1/6 +     5/6
    = 1

Which is a bit fiddly but hardly impossible. I think there is pedagogical value in doing the algebra accurately and I think it is annoying enough that the trick seems useful and memorable when it is pointed out. The trick is also quite broadly applicable to physics problems where there is conservation of some quantity.

It is not impossible but I dont think generally grade 5-6 students have a chance to solve this (of course it depends how much training they received)

Yeah I think the examples given are probably examples of hard exercises or the grades are a poor translation (maybe grade 1 has 6 year olds, for example.)

Another solution is with geometry:

Start with a 1x1x1 cube of coffee. Remove top half and replace with milk. Now remove left third of resulting combined shape (leaving a 2/3 x 1/2 x 1 cuboid of coffee). Now remove front sixth (leaving a 2/3 x 1/2 x 5/6 cuboid). Now drink it all. Now add up the volumes of the shape and write down the answer.

> I think it is ok as a brain teaser, but there will probably be one kid in the class to see it and all the other kids feel dumb or whatever. But I don't think it teaches you anything

These word problems are intended quite precisely as introduction to algebra. They show you the kinds of questions that algebra can solve in a "structural" way, which makes it way easier to grok algebra later on since motivation for the subject has been provided in such depth.

A good review article on the subject: Persson, Ulf and Toom, André: Word Problems in Russian Mathematical Education, available at http://toomandre.com/my-articles/swedish/ULFENG.PDF

Why do you think this is the starting point of a structured introduction?

K-1 means Kindergarten and 1st grade, so 5 year olds and 6 year olds. In the US earlier grades than that are called PK-3 (pre-Kindergarten and 3 year old) and PK-4. I don't think RSM has classes for these levels.

Correct, they don't have pre-school classes. I attended RSM as a student, and even taught there.

> Edit: this might make the kids think you need some 'magical' insights to do math and if they don't see it they are not apt for it, while the opposite might be true.

That's a valid point. However, what's the opposite to having insights? Is that following routines and/or exhaustively exploring the entire problem space (which the first problem in the GP comment seems to teach)?

Teaching those might have higher pedagogical value than conditioning children to find insights (as - at least at first sight - the increase in skill in those is more directly linked to the effort the child invests in learning) However, the von Neumann story in your sibling comment suggests that some people (and so, some children in the class) will perform routines faster than the other children no matter what. Seeing a "shortcut" solution gives a chance to those who are slower at routines to arrive at a solution fast, too.

Moreover, a lot of real-world problems (in academia as well as in business - from my limited experience in both) are exercises in pattern matching and finding shortcuts rather than in an exhaustive exploration of the problem space - and helping children to collect an arsenal of tricks (and more importantly, teaching them to look for insights and patterns by giving them multiple trick-based problems over the years) prepares them to handle those real-world problems.

Yes I definitely felt dumb in my Russian math classes when someone saw a clever trick and I did not. That made me try harder to look for tricks, it's a challenge in the end.

I ended up with a Math degree very later on. So I don't see how feeling dumb harmed me.

PS: the top level professional math is 90% tricks.

#2 relies on understanding how numbers "work", i.e. on having spent time on playing with them.

#5a is a factorization exercise.

In general, Russian approach to the math includes exposing kids to the toolset as well as the theory. This particular case is solved this way, that is solved that way, etc. Keep on it for a while and it tends to develop an intuition for knowing right away which problem is solved with which approach. So, yeah, if there are shortcuts, you use them. If there aren't any, you brute-force it.

> 5a looks dodgy (probably just trying out numbers)

You can find the 4 zeros of the left side by eyeball, and use that to construct the rest of the equation group you need.

It's expected you've seen and solved this type of problem before taking the test. Not that many kids can come up with a working solution strategy on the fly.

"Trying out numbers" seems like the simpler approach to me. You have to multiply four consecutive numbers to get 1680. One of them must be 7 (since 7 divides 1680), one must be 5 or 10 for the same reason. That leaves only 3 possibilities, and you only have to check the middle one (5 x 6 x 7 x 8) since if the result too big or too small, you'll know which of the other two it is.

OK - you weren't guaranteed the solution was an integer or even real, but you should strongly suspect there's a simple answer because you didn't get generic fourth-degree polynomials in your class. Depending on the class, you might be expected to find one, two or all four solutions - but once you have the first one the rest are much easier.

Following the procedure taught in class (and I'm quite sure they didn't teach "make a guess") is safer. When you show the correct procedure and fumble say, and addition, near the end, you'll still get almost full points.

If you go off-road and get the wrong answer, it's up to how much the teacher likes you.

This doesn't line up with all the other problems, which reward having some insight as well as the concrete algebra/arithmetic skills to finish it out.

If anything, your approach sounds more like what's expected in American schools: either you have been taught a foolproof way to solve problems matching Pattern X, or you stare slack jawed at the question paper thinking "I must have been absent the day they covered this question pattern". Exactly the opposite of the kind of thinking described in the article.

Besides, as in another comment of mine on this page, there is no generic method your teacher could have taught you here that always works. (If you think so, please set the right hand side to 1681 and solve that version...). The principal method I was taught for solving cubics was to make a guess. Numerical methods came later.

Ok, only following procedure I also don't like..

When you have the 4 zeros how do you construct solutions?

You'd have to ask the kids who just took that class. I'm too old to remember ;-)

I like your sense of humour

> 2 also seems to rely on a trick

Pretty sure students are supposed to switch( n % 3 ), and solve for all 3 possible values of the remainder.

> probably just trying out numbers

You can't solve a 4-degree polynomial equation by trying out numbers. It has up to 4 solutions, not guaranteed to be real, let alone rational or integer.

> You can't solve a 4-degree polynomial equation by trying out numbers

On the contrary, in practice you can't solve them any other way! Spot a simple factorization or guess a root is your best chance if you don't fancy working through the quartic formula [0], which will take you multiple pages just to write down the first step, and a numeric approximate solution is not acceptable.

[0] https://math.stackexchange.com/a/1135224