You need the two breaks to be on opposite sides of the stick (since if they are on the same side, one piece will be greater than 1/2, so the other pieces can't match its length).
Probability 1/2
But given that they are on opposite sides, you ALSO need them to be not so far apart that the central segment is >1/2.
I.e., wherever the left break is in the left half, the right break needs to be to the left of within the right half.
By symmetry, this has probability 1/2.
Combined probability is 1/2 times 1/2 = 1/4.
Condition 1 is this.
If positions of breaks are X and y, it is ok to have a high X and low y or vice versa.
Condition 2 is this
X and y are not allowed to be more than 0.5 apart. (otherwise the middle piece of stick is more than 0.5, and you can't make a triangle).
Combining the two criteria
Or, when drawn properly by Guy rather than sketched uglily witha thumb on a phone:
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This is why I could never stand for election for the society of X or y technique. And even if I was crazy enough to do so, I would certainly never be elected !
You must publically advocate that the technique is wonderful and amazing.
This Sam Loyd question is generating lots of very nice equation-based answers, but let me try a method, derived from Zugzwang et al, to see if I can do it without so much algebra, just by thinking.
Suppose we CHANGE the question to this:
"The hour hand and minute hand are SUPERIMPOSED on each other. How long till this superimposition next happens?"
Here is my approach. Suppose it is now just-after-two o'clock. When will this next happen?
First answerer is correct.
And this will continue on in a similar pattern.
When you get to "just-after-11 o'clock", what actually is that time, after 11 o'clock, that the hour and minute hands superimpose?