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Triangle From a Ruler

Probability puzzles require you to weigh all the possibilities and pick the most likely outcome.

 Puzzle ID: #26982 Fun: (2.45) Difficulty: (2.8) Category: Probability Submitted By: phrebh Corrected By: phrebh

Fred had a ruler that was exactly 12 inches long. His second cousin, Pop, was practicing with her Samurai sword and made two straight slashes at arbitrary spots on the ruler, cutting it into three pieces. What is the probability that the pieces of Fred's ruler can form a triangle?

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 Miss_Xuni Nov 17, 2005 Meh, Geometry. I did finally figure it out though, which made me happy. smarty_blondy Nov 18, 2005 Great teaser, however eh... i can't help but ask, if you look at your hands... how many fingers do you see? BOOKLOVER1 Nov 18, 2005 boring cdrock Nov 18, 2005 Good one Phrebh, took me a while swimmergirl_mrh Nov 19, 2005 no offence but that is kinda stupid. I mean Who Cares about Freds ruler???? suyolcusu Dec 01, 2005 Cutting a stick into three parts to form a triangle The solution presented is not exactly right, because it assumes the generation of two uniformly distributed random numbers between 0 and 1. The ruler of unit length is then cut at these positions. The probability of forming a triangle is then 0.25 as given in the answer. A more realistic way is to cut the ruler first at location x (x is uniformly distributed between 0 and 1) then cut the remaining piece at location y (y is uniformly distributed between 0 and "1-x"). The resulting three pieces have lengths "x", "y", and "1-x-y". The probability of having a triangle is then P= integral from 0 to 0.5 {integral from(0.5-x) to (0.5) [dy/(1-x)]}dx 0.5 P={-x-ln(1-x} 0 P=0.193147 The same answer can be obtained by simulation. A simple BASIC program using one million trials is shown below: 10 REM "Cutting a ruler into three pieces to form a triangle" 11 N=1000000 12 S=0 13 RANDOMIZE 14 FOR I=1 TO N 15 R1=RND(1) 16 A=R1 17 B=1-R1 18 L1=A 19 L2=B 20 R2=RND(1) 21 C=L2*R2 22 L3=L2-C 23 L2=C 24 IF (L1+L2)>L3 THEN 26 25 GOTO 29 26 IF (L1+L3)>L2 THEN 28 27 GOTO 29 28 IF (L2+L3)>L1 THEN S=S+1 29 NEXT I 30 P=S/N 31 PRINT "Probability ="; P 32 END phrebh Dec 01, 2005 I believe you misread the answer. The second cut is taken from the remainder of the ruler. It's just that its length is shown in relation to the first cut. choptlivva Dec 11, 2005 Oh, my head hurts!!! Good teaser I'm sure, Phreb.... but I wiggled through college Math by the skin of my teeth!! I have no idea how, except that the TA that taught the class liked me.... choptlivva Dec 11, 2005 BTW.... I thought all you needed was three straight lines to form a triangle!! Geez... I feel stupid!!! gaiapeach1 Dec 16, 2005 I don't get it. Do you mean three straight edges, with the inside being a triangle? Or do you have three pieces that fit together in a puzzle-piece manner to create a solid triangle? I'm a bit confues. ChessNut Dec 18, 2005 wow...I think I need to go back to playing Chess...you are doing too much Math for me. great job! sebastian Dec 19, 2005 huh??????????????? SilverMoonBee Jan 21, 2006 To be honest, i never liked probability questions. Yours i did enjoy though. udoboy May 26, 2006 If b>6, you can in no way form a triangle. If b = 6, you form a "straight-line" triangle. It seems to me then that, if the first cut is perfectly even, then 100% of the time you have the straight-line triangle. If not, then the second cut, if made on the shorter length, will never produce a triangle. This should be 50% of the time. If neither of those take place, then the third cut, randomly made on the longer piece, needs to not be further than 6" from either endpoint. The probability of this depends on the length of the piece. Kiroho Sep 05, 2006 if any segment > 6 then triangle is not possible if any segment is exactly = 6 then a triangle is not possible, only a straight line is. if all segments are < 6 inches, a triangle is always possible. So the answer is slightly less than 50% phrebh Sep 05, 2006 No, it isn't. You are ignoring the fact that you have to look at two instances (one for each cut), not just one, so you multiply the two 50% probabilities to arrive at 25% (1/2 * 1/2 = 1/4). Kiroho Sep 08, 2006 I just ran it through Exel and you're right. It is about 25%. opqpop Sep 29, 2010 Hurray for geometric probability! It's very powerful abharath27 May 13, 2013 You are making distinction between the first cut and second cut... So a cut of pieces of size 4,4,4 will be counted twice.. This isn't correct right? The correct answer then should be 12.5%