Pure maths, physics, logic (braingames.ru): non-trade-related brain games - page 38
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Yeah, now it's your turn. I'm off for popcorn.
I don't see the point in continuing.... :) You're counting probabilities incorrectly and you're not even sure....
Here is a simpler problem (for grade 5 - 6). We all know how the face of square and triangle is counted...
Where is the mistake?
You have to memorise the volumes of petrol in the tank after each refueling, and dance around them. What happens when driving from a new arbitrarily standing barrel to the nearest (in the same direction) enough gasoline to the same amount (then the route is driven by the assumption of matinduction, because all the missing amount from each barrel has already been pumped into the new one and thus has already entered the tank) and what happens if there is not enough gasoline (there are a few more cases to consider).
I have a very elegant solution (you can't praise yourself - who else can), no formula or God forbid induction...
But in order to do that, you have to know exactly where the drums are and how much fuel they contain.
I have a very elegant solution (you can't praise yourself - who else can), no formulas and God forbid induction...
But to drive like that, you'll have to know exactly where the barrels are and how much fuel is in them.
Let's think about it.
Manov, you're on fire. I'm getting a beer.
There you go, Alexei comes in and rips everyone apart.
A nim problem (actually I saw that there is a nim when I read the comments for the solvers; the weight is 5 points):
There is a strip divided into N squares, arranged horizontally in a row (N > 3). On the first three squares, counting from the right, there is a chip. Two players play a game in which each turn any piece moves to the left to any empty cell (jumping over other pieces is allowed). Players take turns moving. The one who fails to make another move loses. Who has a winning strategy?
By the way, what haven't we decided yet? Cutting the circle - definitely not solved. A reminder (the weight is only 4):
Cut the circle into several equal (matching when superimposed) parts so that the centre of the circle does not lie on the border of at least one of them.
Another one (3 points):
You have to choose between two cylinders. Externally, the cylinders are exactly the same: they have the same size and weight, each one is painted green. But one inside is hollow and made of gold, the other is solid (without hollows) and made of a non-magnetic alloy. You can't damage the cylinders or scratch the paint. Is it very easy to find out which cylinder is made of gold?
(5 points - I don't understand why):
A megamoggle went into a pet shop and bought two plus half of the remaining rabbits. The second megamoggle bought three plus a third of the remaining rabbits. The third megabrain bought four plus a quarter of the remaining rabbits. And so on, until it was no longer possible to divide the rabbits. How many maximum megamrains could buy rabbits?
Cut the circle into several equal (overlapping) parts so that the centre of the circle does not lie on the border of at least one of them.
Lay out the solution and let's forget it )
You have to choose between two cylinders. Externally the cylinders are exactly the same: they have the same size and weight, each one is painted green. But one inside is hollow and made of gold, the other is solid (without hollows) and made of a non-magnetic alloy. You can't damage the cylinders or scratch the paint. Is it very easy to find out which cylinder is made of gold?
Well this one is easy.
Also mine with a brick and 30m :)
Cut a circle into several equal (overlapping) parts so that the centre of the circle does not lie on the border of at least one of them.
The condition sounds bipartite...
If several parts do not reach the centre of the circle, but other parts do - is it a solution?
That's the only way it works for me.... :(
There is someone else interested.
Also mine with a brick and 30 metres :)
Exactly.
Manov: If a few pieces don't reach the centre of the circle, but other pieces do, is that a solution?
Here's an example with a square:
All parts (triangles) are equal. There are 4 triangles passing through the centre of the square. But let's say the borders of the blue triangles don't go through the centre of the square.
Yes. I have the same thing, only prettier:
I drew it in Paint, too. All the arcs are exactly arcs of circles, not Bézier curves. Explanation for those not in the tank: the radii of all the arcs are equal to the radius of the circle itself.
And it all started with a construction like this:
can we replace two neighbouring barrels with one - total barrel for cases where it will not improve passing?
If there is enough petrol in each of the neighbouring barrels to cover the distance between them, then replace/merge them into one and place it anywhere between these two (or in place of either). In this case, nothing will change for the better as reaching either of these barrels in the previous arrangement automatically meant reaching the other and the total amount of petrol gained was the same.
Some confusion (ambiguity) arises in the marked places. It is solvable, but clarifications are needed. However, instead I have invented a completely equivalent (and transparent) substitution. If each of the neighbouring barrels has enough petrol to cover the distance between them, then replace/drain them into one, cut out the section separating them and pour from the total barrel the amount of petrol needed to cover the cut out section. The swap has now become completely symmetrical. In this variant, by the way, it becomes completely obvious that the passage of the ring is always possible in both directions.
It is also possible to replace two barrels by one, if in one of the neighboring barrels enough gasoline to get to the other - we pour gasoline in it. In this case there is also no improvement for any of the options.
Overall picture:
i.e. by changing option (1) to (2), assuming there is enough petrol in t-C (in litres) for the distance (CB), nothing has changed for all passage options from point D - if reaching t-C, then reaching B as well and having an x+y-BC increase in petrol, as with the new arrangement. On the other hand, when driving from A only the situation is worse - the vehicle might not have enough petrol to reach waypoint C, but if it is enough, the petrol gain will be x+y-AC - the same as before.
And so we drain as long as possible. It is impossible when the distance between any two barrels is greater than the petrol in any of them. But it is impossible because then the total in the barrels would be less than 100l.
So as a result of draining there will be only one barrel with 100l. The one which remains is the starting drum for the original configuration) of the drums.