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#151 Help Me ! » Regular concave polygons » 2013-04-19 20:28:37
- anna_gg
- Replies: 12
What is the regular concave polygon with the smallest number of sides?
All its sides must be equal and all its internal or external angles must be equal as well (i.e. all its internal angles must be either equal or negative, that is, either θ or -θ).
#152 Re: Puzzles and Games » Shared birthdays » 2013-03-30 07:35:15
Thank you! Very useful!
#153 Re: Puzzles and Games » Shared birthdays » 2013-03-30 05:57:52
No I haven't; actually I wrote the solution in a Word doc, but when I copied it here, the formatting was screwed up 
(
#154 Re: Puzzles and Games » Shared birthdays » 2013-03-30 04:58:00
I will try to share my solution in a more explanatory way (Sorry, Bobby, I don't imply that yours is not easy to understand - it is just that I am a novice and not very familiar with complicated solutions!!):
We first must calculate all the possible ways to get 2 months out of 12, that is
Then we must calculate all the different ways by which we can arrange the birthdays of 6 people in these 2 months: Either 5 people have their birthday in the first month and 1 in the second, or 4 in the first and 2 in the second etc. Obviously we do not consider the case of 0/6 or 6/0. For the first case, we first get 1 out of 6 (for the first persons birthday) and then for the second persons it will be 5 out of 5, and so on.
Here is the calculation:
The total probability is the product of the first two (66 x 62) divided by the total number of all different ways by which 6 people can have their birthdays in 12 different months, that is, 12^6.
So we have
#155 Re: Puzzles and Games » Logicians and stamps (again)... » 2013-03-28 10:59:29
Yes they do because they are sitting on a circle, thus they can see each other's forehead.
No they don't know what color are the stamps that the moderator still has.
#156 Puzzles and Games » Logicians and stamps (again)... » 2013-03-28 01:43:29
- anna_gg
- Replies: 2
An unknown number n of logicians will play the following game:
A moderator will stick two stamps on each logicians' forehead. He tells everyone that he has n+1 red stamps and n+1 black stamps. They do not know, however, that the moderator has already stuck one red and one black stamp on the forehead of each logician, except for one to whom he has stuck two red stamps.
The logicians are sitting on a circle so that anyone can see everyone else's stamps. The moderator asks them in turn, starting from the logician who is sitting in the position Nr 1, "what color are your stamps?" The logician with the two red stamps on his forehead is sitting in the position Nr x (unknown to us).
For which values of n and x the logician with the two red stamps can guess the color of his own stamps?
#157 Re: Puzzles and Games » Shared birthdays » 2013-03-28 01:42:08
Right 
#158 Re: Puzzles and Games » Shared birthdays » 2013-03-24 01:28:11
Hi;
See my corrected description.
#159 Puzzles and Games » Shared birthdays » 2013-03-24 00:28:49
- anna_gg
- Replies: 27
What are the chances that 6 people celebrate their Birthday in the same 2 months? Assume all months are equal.
#160 Re: Puzzles and Games » Find the number » 2013-03-24 00:25:52
Sorry guys, I have been etremely busy for the past week...
All of your answers are correct!
Congrats!
#161 Puzzles and Games » Find the number » 2013-03-16 14:07:22
- anna_gg
- Replies: 11
What is the smallest natural number such that its double is a perfect square, its triple is a perfect cube and its quintuple is a perfect quint?
#162 Re: Puzzles and Games » Double and half » 2013-03-05 08:57:44
Excellent! Thanks!
#163 Re: Puzzles and Games » Double and half » 2013-02-28 04:39:16
CORRECT! I found it by using Excel but now am trying to formulate it.
#164 Re: Puzzles and Games » Double and half » 2013-02-26 01:08:24
Here is another variation: Find the smallest natural number for which, if you move its last digit at the beginning, you get a number that is 5 times the original.
#165 Re: Puzzles and Games » Another variation to the two envelopes puzzle » 2013-02-02 23:37:00
The correct answer is the one that muxdemux submitted but I am not sure about the reasoning for B. Why wouldn't he want to swap?
#166 Re: Puzzles and Games » Another variation to the two envelopes puzzle » 2013-02-02 22:41:39
For any amount M that A receives, there's 50%-50% probability that B gets either 2M or M/2, so on average A would gain from swapping and by symmetry B would lose. But what if we apply the same logic starting from B? How does the reference to B's amount ($100) change our reasoning?
#167 Puzzles and Games » Another variation to the two envelopes puzzle » 2013-01-20 03:49:41
- anna_gg
- Replies: 5
Two players, A and B, are going to play a game. A perfect logician explains the terms:
I have several envelopes containing different amounts of money. I will randomly pick one of them, see the amount that it contains and will give it closed to player A. Then I toss a coin and if I get tails, I will get an empty envelope and put half the amount of player's A envelope, while if I get heads, will put double. I will then give this envelope (closed) to player B.
Then I will invite each of you privately and ask you to decide whether you will swap envelopes or not. If you both agree in swapping, you will do so, otherwise you will keep your initial envelopes.
A and B agree with the procedure and then A asks B to reveal his amount, so that they get an idea on what to propose to the logician. They both see that B has $100. Right afterwards, each of them must meet the logician in private to announce their decision. Which decision ensures the biggest expected gain for players A and B separately? Explain your answer.
#168 Re: Puzzles and Games » Double and half » 2012-12-02 05:34:34
Hi;
Great! It is said that there are infinitely many solutions. I have used Excel for the calculations but have not been able to find any at the range 1-5,000,000. Then I gave up
#169 Re: Puzzles and Games » Double and half » 2012-12-02 03:19:55
Hi anna_gg;
Bobbym,
Sorry, I had made a mistake - please read the new description!
#170 Puzzles and Games » Double and half » 2012-12-02 01:55:19
- anna_gg
- Replies: 12
Find a natural number for which, if you move its first digit at the end, you get a number that is half the original one
(e.g. 81345--->13458 but the resulting number must be the half of the original).
#171 Re: Help Me ! » Planes Of Symmetry » 2012-11-12 01:16:57
Yessss!! And there are only 3 such planes, because for every two sets of sides we get the same plane.
Thanks Stefy!!
#172 Re: Help Me ! » Planes Of Symmetry » 2012-11-11 23:16:33
Nope! There are three more, one for each pair of midpoints of opposite edges of the tetrahedron.
So 4 + 3 = 7 such planes?
Can you make a drawing for the 3 last planes?
Thanks!!
#173 Re: Help Me ! » Planes Of Symmetry » 2012-11-11 21:03:05
Bob, I think this is it, although I do not understand your second drawing.
So I guess we have four such planes, correct?
Thanks,
Katerina
#174 Re: Help Me ! » Planes Of Symmetry » 2012-11-11 05:11:41
hi anna_gg
I have, of course, assumed that the triangular pyramid is regular; ie. all its faces are equilateral triangles.
If it is 'sitting' on one face with one vertex in the air, then, a line drawn straight down from that vertex to the base, will go through the centre of gravity (centroid) of the base. Let's call that line (don't know the proper term for it) an axis. Turn the pyramid so that a different vertex is on top and repeat the construction. This new axis will intersect the first. You can do this four times altogether and get four axes, all intersecting at the same point. This point is the centre of gravity of the pyramid and it is the same distance from all four vertices.
Furthermore it is the only point that is the same distance from all four vertices. But you want a vertical plane.
Ok let's try this. I'll find a line in the base (ABC) that is the same distance from all three base vertices. Then make a vertical plane going upwards from this line. See picture. The base is ABC. The line is shown dotted. It's distance is the same from all three base vertices.
This plane is, therefore, (i) vertical (ii) the same distance from all three base vertices.
Now suppose we are looking straight down from above on the pyramid. Let D be the last vertex. You can see it will be much closer to the plane.
Conclusion. I think it is not possible to create the plane you want.
Bob
Just to clarify, I did not say that the plane must be vertical. I only clarified that the (vertical) distances of each vertex from this plane must be equal. Sorry for the confusion!
#175 Re: Help Me ! » Planes Of Symmetry » 2012-11-11 00:04:02
hi anna_gg
The picture I suggested goes through two vertices and the midpoint of the side joining the other vertices.
So it is the same distance from only two of the vertices, and it is a plane of symmetry.
There is one point only that is the same distance from all four vertices, (see picture below), so you could draw any plane that all that goes through this point. It would not be a plane of symmetry.
Bob
Hi Bob,
Actually we are looking for a plane that has equal (vertical, obviously) distance from ALL FOUR vertices, so I guess this plane must not pass from any of the sides (because in this case, for the 2 of them, distance will be zero).
Anna