Pure maths, physics, logic (braingames.ru): non-trade-related brain games - page 115
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With balls - or with carts?
We make an equation for the sloth based on dp/dt = m(t)dv/dt + vdm/dt = -mu m(t) g. That is, we reveal the momentum explicitly.
Draw up an equation for the worker, taking into account both forces acting on the cart.
We note their almost complete similarity.
And make it complete by multiplying the equation for the worker by the integrating factor equal to 1 at zero.
It turns out that the new equation for the worker can be interpreted as follows: the former worker now does not dump snow, but also lies on the cart and does nothing. But the snow increases the mass of the cart according to a different law - not linear, but exponential. Further the proof is obvious, given that the integrating factor is an exponent equal to 1 at zero andgreater than a linear function.
Multiabookafniassil.
Don't cheat, say it straightforwardly: so which cart goes further?
;-)
Despicable invaders have taken over a village of megabrains, lined them up one after another in a column so that each successive one sees all the previous ones. Each megabrain is hooded in black or white, so that no megabrain can see his own hood. Starting with the very last one (the one who sees everyone but himself), each megabrain is asked the colour of his cap in turn. If he's wrong, he gets killed, but just in case, the megabrains agreed in advance how to minimize the number of people killed. What did the megabrains agree on?
I've come up with such a strategy:
The strategy consists of two parts (sub-strategies)
I. Regular substrategy// With it, megabrains start if there is an opportunity, i.e., if the back sees a certain pattern
1) If a back megabrain unambiguously sees a certain regularity in the arrangement of colours ahead of it, it follows this regularity when calculating its own colour, which it names.
// it guarantees survival if it is right, and in any case notifies the next one that he is within the pattern, ensuring 100% survival.
This sub-strategy is followed by megabrains as long as the pattern can be detected (there must be enough "patterns" ahead of the observer to do so).
After a pattern can no longer be detected (i.e. if any megamotle in the chain (and hence all following ones) can no longer see any pattern, it moves on to the second sub-strategy.
II. irregular sub-strategy// The sub-strategy proceeds from the assumption of random alternation of colours
1) The rear megamograin has no chance of recognizing its colour, so it names the colour of the one in front of it. It survives with a probability of 1/2
2) The next one calls the colour that he has heard from the rear one // 100% survival
3) Next acts as a posterior-as-dan-strategy => (1)
--
By following these two sub-strategies in the worst case (when the first sub-strategy is not possible) - 3/4 of the population survives. not bad at all.
--
I'm sure it's impossible to get a higher survival rate, although there may be variations of the agreements.
MD: Не юли, скажи прямо: так какая телега дальше проедет?
Well, I told you not to say the decision!
There is a strategy where only one MM has a 50% chance of being killed. Simple as hell if you ask me :)
I don't believe it.
I thought so too at first - until I checked all the layouts.
// For the optimists: to survive, it's not enough to know your colour, you also need to name it.
Lazy.
Of course, no one believes it until they see the solution. By the way, there is a similar problem, which for some reason has more weight:
(4) The sneaky invaders didn't like the fact that they killed very few people in the village of megabrains, so they decided to complicate the task. They again put the megamogs in a column behind each other so that each successive one could see all the previous ones. But this time they took hoods of seven colours (red, orange, yellow, green, blue, blue, purple), put them on the megamogs so that each megamog can't see his own hood. Starting with the very last one (the one who sees everyone but himself), each megabrain is asked the colour of his cap in turn. If he is wrong, he is killed. But as always, the megabrains agree in advance on how to minimise the number of people killed. What did the mega-brains agree on?
TheXpert: Yay :)
If you see my solution, you'll be jealous...
You'll see my solution - you'll be envious...
No way I'm admitting mine is worse :) I agree on equality. And yours is basically clear from here --
But the snow increases the mass of the cart according to a different law - not a linear one, but an exponential one.
But I'd like to see it. I'm not good at diffusers.
OK, here it is:
There's a division mark missing in one place (after 'from'). Click on the picture, it will show better.
By the way, the reasoning is modified in case the snow initially falls unevenly.
P.S. I realise that the solution is not elementary. But it changes the reality!
OK, here it is:
There's a division mark missing in one place (after 'from'). Click on the picture, it will show better.
By the way, the reasoning is modified in case the snow initially falls unevenly.
P.S. I realise that the solution is not elementary. But it changes the reality!