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#337727 Orderly numbers
Posted by bonanova on 07 March 2014  03:00 PM
#337617 Triangles inside circles
Posted by bonanova on 28 February 2014  12:18 AM
A few puzzles posted in this forum have related to random triangles inside a circle.
By evaluating nasty integrals, or by my preferred method, simulation, it can be shown, perhaps surprisingly, that triangles constructed from sets of three uniformly chosen points within a circle cover only about 7.388% of the circle's area on average. After looking at Phil's recent problem on the subject, I simulated 1 million triangles to determine the median area. It turns out to be about 5.335% of the circle's area. Read: a random triangle has a 50% chance of being smaller.
If the distribution of randomtriangle areas has a mean of about 7.4% and a median of about 5.3%, what value might you expect for the mode?
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#337573 Orderly numbers
Posted by bonanova on 27 February 2014  08:48 AM
Consider the numbers from one to one million: 1, 2, 3, ..., 999998, 999999, 1000000.
What is remarkable about the numbers 40, 8, and 2202?
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#337517 The kindest (shortest) cut of all
Posted by bonanova on 25 February 2014  10:39 PM
Here is a piece of plywood in the shape of an isosceles triangle.
The side lengths are 1, 1, sqrt(2) units.
Quick and dirty representation:
A
 \
 \
 \
 \
BC
The angle at B is a right angle.
We'd like to cut this into two pieces of equal area.
There are many ways to do this with a single cut.
Which cut has the shortest distance?
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#337512 Making 271
Posted by bonanova on 25 February 2014  09:13 PM
Is this with regards or without regards to orderings of the summands?
I would guess "ways" means combinations.
The permutations of 271 1's are not that interesting.
 1
#337443 Binary tictactoe
Posted by bonanova on 20 February 2014  11:07 AM
In the alldigital future, X and O are banished from the game of tictactoe.
They are replaced by 1 and 0, the the result of such a game might look like this:
1  0  1
 +  + 
0  1  1
 +  + 
0  1  0
Under the usual rules that require getting 3inarow, it would be a draw.
But this is the digital age, and there are different rules for winning.
If we sum the eight rows of three numbers we get 2, 2, 1 (horizontally) 1, 2, 2 (vertically) and 2, 2 (diagonally).
Six of the sums are even, and two are odd.
The final parity of the board is thus even, and the game is said to have an even outcome.
If there were more odd sums than even, the game would have an odd outcome.
If there were four even (and therefore four odd) sums, the game would have a neutral outcome.
The game is played as follows:
The winner of a faircoin toss (call him player A) chooses whether to play first or second.
The other player (call her player B) decides whether she wants an odd, even, or neutral game outcome.
On each turn, a player places his choice of either a 1 or a 0 on any unoccupied place on the grid.
As in normal tictactoe, players alternate turns; but here on each turn a player may play either a 0 or a 1.
When the places are filled, the board is examined to determine whether it is odd, even or neutral.
If the final board parity matches player B's choice, player B wins; otherwise player A wins.
The questions to answer are:
 Is there an advantage to winning the coin toss?
 Is there a winning strategy for either player?
 1
#337442 The clock in the mirror
Posted by bonanova on 20 February 2014  10:07 AM
A while ago, BMAD posted a problem about a clock with indistinguishable hour and minute hands and asked at what times of day, between the hours of noon and midnight, it was impossible to unambiguously determine the time. The hands moved continuously.
This puzzle asks a related question. At what times of day, between the hours of noon and midnight, is it impossible to distinguish the hands of such a clock from those of its mirror image? Clearly noon is one of these times, but not in general thereafter  since the clock's hands will move clockwise while the hands of its mirror image will move counterclockwise.
 1
#337283 Betting on red
Posted by bonanova on 30 January 2014  02:46 PM
Here's a variation on a previous puzzle that let you make multiple bets while choosing the color of the next card.
I shuffle an ordinary deck of playing cards and then turn over the top card sequentially so that you can see it. At any time you may ask me to stop and place a $1 bet that the next card to be exposed will be red. If you never ask me to stop, you will automatically bet on the last card. To summarize:
 You can bet on only one card.
 You don't get to choose the color.
 You must bet the card will be red.
What is your best strategy?
How much better than even can you do?
 1
#337125 Second 2014 puzzle. Dissection of a square
Posted by bonanova on 01 January 2014  11:13 PM
From P draw lines to the vertices and perpendiculars to the sides.
This defines eight triangles that meet at P.
Ignoring permutation of identical pieces, how many other ways can these triangles form a square?
 1
#337062 Word play. Because they're fun to play with
Posted by bonanova on 21 December 2013  11:55 PM
 1
#337061 Pun alert  II. Somebody stop me!
Posted by bonanova on 21 December 2013  11:49 PM
Moe: Did you hear about the actress who stabbed her boyfriend?
Joe: No, I didn't.
Moe: Yeah, it was, uh, Reese ...
Joe: Witherspoon?
Moe: No, with a knife.
 1
#337010 Infinite Flips
Posted by bonanova on 16 December 2013  01:02 AM
 1
#333964 3 x 4 ConnectTheDots Game
Posted by bonanova on 08 July 2013  05:57 PM
This debate about touching vs crossing was the heated debate for which I spoke of earlier :)The difference in this problem's contest is subtle. Think of a drawn line as a physical object with defined spaced (which is key to understanding how my homeland uses the words). If one draws a line and touches a line they simply stood in the other line's space like standing on a doorstep (the drawn line) to knock on a door and left if no one answers, while If one draws a line that crosses the line they knocked on the door and went inside.So I think key to solving this problem is understanding that it treats a line segment as an object with physical properties, area, length, etc. and not as a cutting tool.I believe, one of my friends best described this idea as an old Tron game where two people cut off the board drawing segments in motorbikes.I don't think that works. We are saying that the green curve below, connecting two green dots,by virtue of sharing a single point (touching but not crossing)isolates the lower red dot from the upper red dot. (The green characters do not actually touch, they can't, but they represent touching lines.) aO dOfsO a\ fs/ sa\e / asd\/ asd/\ as/df\ s/adfa\ /cxvcxz\ \cxvOcz/ a\s/f\/ s/\af/ /cx\_/ Touching and crossing cannot be distinguished, once the line has been drawn.It's exactly the same locus of points. But let's say it's all about how the lineswere drawn, not how they end up. a line cannot be drawn in a manner bywhich an existing line is crossed by the pencil. The "touching but not crossing" line still does not isolate the red dots. The red dots can still be connected by a red line that also sharesthe common touching point without crossing either the left part or the right part of the green line. Thus: aO dOfsO a\dsfs/ sa\e / asd\/ asd/\ as/df\ s/adfa\ /cxvcxz\ \cxvOcz/ a\s/f\/ s/\af/ /cx\_/ So I still contend you can isolate a portion of the plane, for the purposes of this game,but only by using a line segment of infinite extent or by crossing an existing line.
The red line just stands on the doorstep as well.
 1
#332917 Probability of a rope cut exactly in half, in a single slice
Posted by bonanova on 11 June 2013  05:09 AM
 2
#332708 A stand up Logician
Posted by bonanova on 30 May 2013  05:30 AM
Spoiler for Slight change in the process does itHave them stand in a straight line.Instead of requesting a show of hands, ask them to take a step forward. Liars will step forward at one or the other request. Truthtellers from out of town will take two steps.Truthtelling locals wont' move.The onesteppers comprise all of the liars.
Truthtellers (T) and Liars (L) who are Residents ( R) or from Out of town (O).
"All from out of town take a step forward." (TO and LR step forward one step.)
"All who stepped forward take a step forward." (TO and LO step forward one step.)
Those who took one step are LR and LO: only liars and all the liars.
You have thus used the same method to distinguish the Ls from the Ts.
Q.E.D.
 1
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