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Puzzling Travels of a Tom-Cat. Answered

Here's a puzzle I came across recently.  I've changed the context to stop people googling the answer :-

A big ginger tom-cat leaves his home at 8:00 one morning and goes trundling around the neighbourhood, pausing here and there to 'leave his calling-card', stopping for a brief nap or two in a particularly warm spot, walking along fences, running through unfriendly gardens (woof woof) and  finding a warm car bonnet in a garage at 6:00 in the evening for a rest.  Unfortunately, the owner shuts the garage door and the cat is trapped for the night.

At 8:00 the next morning the garage door opens and the cat comes scooting out.  Over the course of the day he takes the exact same path he took the day before but in reverse.  He stops in different places, makes his mark in other places and takes his naps at different places along the way, arriving back at his own house at 6:00 in the evening for a well deserved meal.

The question :-
What is the likelihood that there will be a time and place along the route where he is at EXACTLY the same point on the route at EXACTLY the same time of day on both days?


Ah, but, what if... the cat's route goes across the international date line? (Probably in Vanatu or on a aircraft carrier or something.) Or his first journey is on the Saturday the clock changes and the return is on the Sunday. Since these factors have not been specifically ruled out then there is a statistical possibility that they could be when the cat is travelling. The clocks change twice a year (spring forward, fall back) so it has to be taken into account.

The puzzle is somewhat artificial anyway, as cats only know one time - Time to Eat, with minor subdivisions such as 'i'll tolerate you petting me for a while as I do actually quite like it', 'I think I'll go and chase something' and 'time for a nap'.  Tom-cats have another fairly important time too!    (And on aircraft carriers, they're called 'hangars' #;¬)

There's no location on land, excluding Antarctica, which includes the international date line. Just so you know. And clock time isn't solar time :-)

Dr. Knotter's right, of course. And there's a lovely diagram in Tufte's Visual Display of Quantitative Information (cover and p 31) showing a similar effect. Trains going in opposite directions along the same route will always pass each other somewhere, even if it's not at the midpoint.

As long as the cat takes the EXACT same path on the way back, there will ALWAYS be a spot along it's path where it is at the same spot at the same time.

Imagine plotting a graph of the distance of the cat from its house as a function of time. On the first day, the line goes from the bottom left to the top right, and on the second day, it goes from the top left to the bottom right. Since the cat can't suddenly teleport farther along it's path (unless it's Schroedinger's cat!) the two lines must cross at some point, which corresponds to being at the same position at the same time of day.

It makes me happy to use my mathematics background every once in awhile. :)

Yep, I found it counter-intuitive when I first thought about it because of the intentional confusion of the random speeds and times, but then I looked at it from another angle and imagined the cat simultaneously doing both journeys on the same day - more a Heisenberg cat, this one.  At some point he must 'meet himself', i.e. your crossing point, and he is then at the same time and place.
The original riddle I saw related to a monk on a pilgrimage up and down a mountain.