[Archive!] Pure mathematics, physics, chemistry, etc.: brain-training problems not related to trade in any way - page 169
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Вопрос на засыпку, тем кто не спит: Что это такое и зачем оно нужно?
Правильный ответ - завтра.
Four-phase plate barbambulator for putting polar bears to sleep
You scared the Matemata off.
Does anyone have any thoughts on a systematic search for cube markup options? Or let's put it aside until we have something serious on it? Basically, the problem is solved formally with the frontal method, 24 solutions are obtained. What can be obtained from them by some symmetry transformations, it is not very clear yet.
P.S. Here is a simple problem: A circle is divided by radii into 6 equal sectors. Each sector contains a chip. It is allowed to move any two chips to adjacent sectors simultaneously: one clockwise and the other counterclockwise. Is it possible to collect all the chips in one sector in this way?
Does anyone have any thoughts on a systematic search for cube markup options? Or let's put it aside until something serious is done on it? Basically, the problem is solved formally with the frontal method, 24 solutions are obtained. What can be obtained from them by some symmetry transformations, it is not very clear yet.
P.S. Here is a simple problem: A circle is divided by radii into 6 equal sectors. Each sector contains a chip. It is allowed to move any two chips to adjacent sectors simultaneously: one clockwise and the other counterclockwise. Is it possible to collect all the chips in one sector in this way?
I guess not, the chips will be collected in 2 sectors. But, I think I have a catch :)
alsu, a big request, don't post the solution. I think you solved it a long time ago.
Richie, do you want to feel the joy of solving a boring math problem - even with a few hints?
P.S. Okay, Richie is probably asleep by now. We'll decide who's interested and who's still awake.
alsu, большая просьба, не выкладывай решение. Думаю, ты ее давно решил.
Richie, хочешь почувствовать радость решения скучной математической задачки - пусть даже с небольшими подсказками?
P.S. Ладно, Richie уже спит, наверно. Будем решать, кому интересно и кто не спит еще.
You can mark two chips in adjacent sectors as "A" and "B" and try to bring them together into one sector in this way.
The distance between the chips is 5 sectors (in one direction, we won't consider the other for space, but it is odd there too), in one move we change the distance to an even value, or to 0. The problem has no solution.
Можно пометить две фишкив соседних секторах как "А" и "В" и попробовать их свести таким образом в один сектор.
Расстояние между фишками 5 секторов (в одном направлении, другое для просторы рассматривать не будем, но там тоже нечетное), за один ход мы изменяем расстояние на четное значение, либо на 0. видим, что фишки в одном секторе никогда не окажутся. Задача решения не имеет.
a. there's another possibility there - two chips in opposite sectors. But the result is the same.
У кого-нить появились мысли о систематическом поиске вариантов разметки куба? Или ну ее нафиг, отложим в долгий ящик, пока по ней не появится что серьезное? В принципе формально задача решена лобовым методом, получены 24 решения. Что можно получить из них какими-нибудь преобразованиями симметрии, пока не очень ясно.
I suggest that we forget it.
This type of problem cannot have an elegant simple solution a priori. The most elegant solution is to use branch and boundary method instead of brute force. But since the problem is solved, there's no point.
It is impossible to collect all the chips in one sector.
It could be simpler, vegetate: mark the chips with numbers according to the sector number, from 1 to 6. On the first move (one clockwise, the second counter-clockwise) the chips will change numbers, but their sum is invariant, i.e. always equal to 21. So, if they are all in the same sector, then 21 is a multiple of 6. Contradiction.
Whoever solves the problem and proves its solution, can consider himself a cool mathematician.
For three circles of arbitrary radius find a triangle of maximal area inscribed into the shaded figure.
But this is so -- if you have plenty of free time and ambition and the desire to break your brain.
Are the circles arranged exactly like this and not otherwise?