Pure maths, physics, logic (braingames.ru): non-trade-related brain games - page 212
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About division of the trapezium bases.
I will not prove it, I will show and explain. If you understand the logic, it is not difficult to prove it.
A successful simplification helped me to understand it. Consider a degenerate version of a trapezium: a trapezium with parallel sides - a parallelogram. Formally, there is no intersection point of its sides, but lines parallel to sides of a parallelogram are equivalent to rays coming from this point. For maximum clarity, let us make it a rectangle as well :)
So, let us have a look at the following picture:
This picture demonstrates "the effect of addition of spatial frequencies" arising at intersections of diagonal lines drawn inside the rectangle. One can see how having as initial reference points only the points which divide the rectangle basis into 4 parts, we can divide it into 3, into 5, into 6 and into 12 equal parts, using intersections of "fractional diagonals" and vertical lines drawn through these intersection points as a division means.It seems to me that this picture makes things so clear, that no other explanation is needed. It remains only to state that the principle remains valid for any parallelogram, and also for any trapezium. In the case of trapezoids, the rays drawn from the intersection of the extensions of the sides should be used as a substitute for the vertical lines:
// In this case, the division of the bases into 5 equal parts is illustrated.
We can also add that the horizontal lines drawn through the same intersection points divide the sides of the rectangle (or parallelogram) into equal parts (and by the same amount):
As for the corresponding horizontal lines in a trapezoid, the division there is unequal, and more interesting. The curious may try to work out for themselves the resulting relations between the parts:
--
It seems to me that the given pictures fully clarify the work and correctness of the generator
With this principle in hand, it is not very difficult to divide the base of any parallelogram, rectangle or trapezium in any rational ratio. The same method can easily be adapted to a similar division of the sides of a triangle, given that it can be turned into a trapezium by drawing an auxiliary line parallel to the side of interest.
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Pure Math, Physics, Logic (braingames.ru): Tasks for brains, not related to trading
Mathemat, 2014.07.08 02:16
Yes, it's beautiful. But I don't understand why it's an exact algorithm yet.
I'm thinking of a proof.
Any options?:)
Any options? :)
Pour, turn over, measure the level, turn over again and measure. Then count it on a piece of paper.
--
I've been working on the trapezoidal division at my leisure. I'll let you know.
The ruler is such that it can only connect two points in the plane, at any rate without divisions, as in the trapezium problem )
Bullshit, it says it has divisions :)
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Pure Math, Physics, Logic (braingames.ru): Problems for the brain, not related to trading
Mathemat, 2014.07.06 19:29
Another one, quite practical.
The terror of the Megabrain village by the damned invaders continues. This time, having caught Megamogg, the occupiers gave him a plain full bottle of water and a carbon ruler, demanding that he count the volume of the bottle, otherwise death. Megamraz examined the bottle carefully: it was not shaped, flat, flat-bottomed, with no label. He performed a few actions and gave an answer. How had he managed it?
Weight - 3.
FAQ:
- What an angle piece is, I hope it's clear to most people. It is a ruler in the form of a right triangle with divisions on the cathetuses,
- the walls of the bottle are very thin, so you can ignore the volume,
- the bottle comes with an airtight cap (such as a cork),
- Initially, the bottle is filled to the brim with water. The water can be poured out, but the water cannot be used again,
- the neck of the bottle can have arbitrary very nasty shape - like this (this is my drawing of the whole bottle in my own solution of the problem):
Any options?:)
Any options?:)
and what's difficult about it looks the same as the first apple, as the multiplier is only one apple,
It would be a lot harder to imagine if the second apple was green. )
Bullshit, it says it's with divisions :)
You have two thin-walled opaque cube-shaped vessels (without a top edge) with capacities of 4.096 and 8 litres on the table in front of you. With an unlimited supply of water, how can you quickly measure out exactly 5 litres?
The task is here. The weight of the problem is 5.
FAQ:
- the walls are very thin, their volume is negligible.
- 4.096 is four whole and ninety-six thousandths of a litre, exactly. Exactly 5 litres is exactly 5, not, say, 5.002 litres.
- Opacity means that you cannot, for example, put a smaller cube into a bigger one and pour water into the bigger one up to the edges of the smaller one. Because of the opacity, it cannot be done accurately enough.
- fast is really fast, quite fast. The ten-step decision will not be taken. It is too long.