Last week’s puzzle was to try and find the train in the Nullarbor Plain, given that we know the direction of a train whistle we just heard. A couple of idealizations to make this truly mathematical: the train is a point; the track is a straight line; you notice if you are standing on the track but otherwise you can’t see it. First, realize that there is no way to guarantee you’ll find the train. After all, even if you are standing on the track when you hear the whistle, the chances are only 50-50 of being picked up (the train may be going away from you).
What about going toward the whistle? That turns out to be a worst-case strategy since that is one place you can guarantee the train is never going to be again (unless the train is stationary or you are already on the track).
What about going directly away from the whistle? Again that is another worst-case strategy, guaranteed never to find the train (unless you are already on the track).
The best strategy, although it seems counter-intuitive at first, is to set off at 90 degrees from the direction of the whistle (either way is fine, it’s symmetric). If you find the track then wait. Maybe you chose the wrong direction and you never find the track, maybe the train’s already passed, maybe it was already going away from you when you heard it. But maybe you find the track and you get there before the train and it comes along and picks you up. This strategy maximizes that chance.
Today’s puzzle is another one that seems to have too little information until you think it through:
Both Lisa and I live on a street with house-numbers from 1 to 100. Lisa wanted to know at which number I live.
She asked me: "Is your number larger than 50?"
That seemed to make it a bit too easy so I lied.
Lisa next asked: "Is your number a multiple of 4?"
I was still in a bad mood so I lied again.
Then Lisa asked: "Is your number a square?"
I was feeling a bit guilty about all the lying so this time I told the truth.
Upon this, Lisa said: "I know your number if you tell me whether the first digit is a 3."
Well, I forget whether I lied or not this time. Lisa then told me where I lived. Of course she was completely wrong.
What house number do I actually live at?
Answer next week