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Comment by dhosek | Hacker News Reader

One of my favorites is one that you should be able to do in your head: The product of two numbers is 37, their sum is 18. What is the sum of their reciprocals?

(I happened to encounter this two times in close succession when I was getting my teaching credential: first in a teaching manual and then a day or two later, a couple teachers at the school where I was doing my student teaching were puzzling over it and thought they’d challenge me with it and I gave them the answer immediately which shocked them since they’d spent a long time on solving this with algebra and I did it in my head in less than a second. To be honest, I probably wouldn’t have been so quick at the solution without having already seen it.)

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37 is prime. Are you sure this problem statement is correct?

The problem does not state that the numbers have to be integers. a and b happen to be 9 +- 2 sqrt(11)

>The problem does not state that the numbers have to be integers. a and b happen to be 9 +- 2 sqrt(11)

but the problem does state that you should be able to do it in your head. who exactly should be able to formulate and reduce simultaneous equations in xy then apply the quadratic formula (with some spicy +/- to keep track of) to get an answer with an irrational number, all in their head? usually, when a problem like this is given there is a shortcut that leads to a simple, not only rational but integer, answer.

the statement "you can do it in your head" generally does not entail this much complexity, as the person who said "you can do it in your head" comes out and says after previously spending a fair amount of time working on it.

words matter, people, that's why I didn't throw in the adjective integral even though I could have.

You _can_ literally do this in your head, and also, it doesn't matter what the numbers are, what the product is or what the sum is.

Well, I had to write it down, but I have to write down everything these days. But from the way the problem was phrased, it was obvious you don;t have to actually find to numbers.

You don’t have to do all that.

If you have a+b and a-b you’ll get 2a when added together.

So knowing just the sum we can say that a is 9 in this setup.

Now we need to figure out b.

Multiplying out those you get

a^2 + ab -ab - b^2

And I get a longing for not having started this a phone.

Cancels and fill in what we know and we get 81 - b^2 = 37

b = sqrt(44) = sqrt(4)*sqrt(11) = 2sqrt(11)

None of this is required for solving the problem in your head. All that is required is the ability to add 1-digit unit fractions in your head, as the problem requests.

> the statement "you can do it in your head" generally does not entail this much complexity

It's funny that you jump to accusing OP of falsely claiming you can do it in your head, without apparently considering the alternative: that the intended solution is a simpler one than you outlined.

Trust me, you can do this in your head if you know basic high school level math, and you don't need to solve quadratic equations or keep a ton of numbers in your head at the same time.

If I ask you if 123456789 is a prime number, do you complain that it's not fair to make you perform division on such a long number?

>Trust me, you can do this in your head if you know basic high school level math

yeah, i guess it was a mistake to graduate from MIT undergrad and grad school in quant fields, i should have just stuck with high school math

>If I ask you if 123456789 is a prime number, do you complain that it's not fair to make you perform division on such a long number?

you tell me, is 13717421 prime?

The difference between the two is that it’s clear that 123456789 can’t be prime since the sum of the digits is a multiple of 3, which doesn’t even require finding the sum since we know 1+8, 2+7 up to 4+5 are multiples of 3. I can even tell you that 43717421 isn’t prime without having to do a divisibility test on it by looking at the digits, although it is a bit more tedious than the 123456789 field.

the difference between the two is that I removed the factors of 3 from 123456789 to get 13717421. so much for your secret knowledge of a hyperspecific case.

You're still missing the point of these problems, which is to challenge you to come up with a clever proof rather than brute-force the solution.

dhosek understood the assignment by making an argument that 123456789 is composite without relying on explicit division of a 9-digit number, which most people would find rather difficult to do in their heads.

Similarly, the posted link is about tiling a mutilated chessboard with dominos. Tiling problems in general are NP-hard, so clearly this isn't something you can solve in your head _in general_, but the charm of that specific problem is that you _can_ solve it by making an insightful observation to avoid the brute force computations.

Similarly, for the puzzle you complained about: we are asked to find 1/a + 1/b where a × b = 37 and a + b = 18. The general solution is to solve a system of two linear equations which involves solving a quadratic equation, which is possible, but tedious and difficult to keep in your head, but the entire point of the question is that there is a better way to figure out the result.

That's only 9 digits. Determining if 12345678910 is primes would be outrageous. That's got more digits than I have fingers!

ab = 37; a+b = 18; 1/a + 1/b = b/ab + a/ab = (a+b)/ab = 18/37

Thanks! The mention that it was solved in under a second must have thrown me :-)

> they’d spent a long time on solving this with algebra

I don't get it. I don't see why / how it would take any longer than a second or two to solve 'with algebra'. What does that even mean? You would just maybe write down the steps rather than doing them in your head. Is there any other way to solve the problem?

I'm pretty sure he means they did the problem by first figuring out the numbers a and b. That's the slow way. I can reveal the fast way if you want, but maybe you should think about it a bit more first! :)

I know the 'fast way'. Took me a second or two to get the answer, but it is so obvious that I cannot imagine anyone trying to solve this by calculating a and b first.

For novice students of algebra first learning to solve for values of variables, the idea of NOT solving for the values of the variables is a major step.

Or for high school teachers of algebra as it turned out.

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