The WormHole Billiard Ball Paradox.

The following is a solution that addresses all the points raised by Professor Paul Davies in his piece "The mother of all paradoxes"...

as published in the "Scientific American" special on TIME, volume 21 # 1, page 28

"A Matter of Time It begins, it ends, it's real, it's an illusion. It's the ultimate paradox." S.A
 "The mother of all paradoxes"...

The paradox described by professor Davies considers the scenario of a billiard ball entering a wormhole such that it exits the worm-hole both 'in the temporal past' and also, at a location such that the billiard ball could impact, and deflect, itself preventing it from ever entering the worm hole in the first place.

This possibility is considered a paradox, and according to wikipedia - A paradox is a statement that apparently contradicts itself and yet might be true.[1][2]Most logical paradoxes are known to be invalid arguments but are still valuable in promoting critical thinking.[3]
wikipedia goes on to say "Some paradoxes have revealed errors in definitions assumed to be rigorous"

The critical thinking that I believe shows the definition of this apparent paradox to be invalid, and how the situation may be better understood, and solvable 'timelessly, is as follows....

First we consider the ball entering the wormhole, and appearing from the exit, in the 'temporal past', and aimed at itself.

in the 'time' based view this would seem to be possible, if Einsteins Relativity is said to be not only about 'change' but also about a thing called time, which has a past and a future.

given the standard view of Relativity, it seems possible that the ball could exit the wormhole such that it can deflect itself creating a paradox, which some suggest leads to the need for 'infinite parallel universes' in which all outcomes of all events are possible to avoid such logical clashes of events.

Such a call for 'infinite parallel universe's is far fetched* ... and  can be avoided at a stroke if we consider that all objects just exist 'now', no matter how distorted in shape or rates of change.

*( given that absolutely no proof or reason is given for such conjecture, other than that it 'might resolve a suggested 'paradox' '- logically, the loosest grounds indeed for suggesting something, and if scientists think about it, very bad circular logic)

While most people seem to assume otherwise, a careful examination of Einsteins paper 'On the electrodynamics of moving bodies' shows that his workings do not prove the existence of time, but only assumes 'times' existence.

Critically, in section 1 of 'Electrodynamics', Einstein outlines 'time' by saying....

If, for instance, I say, “That train arrives here at 7 o'clock,” I mean something like this: “The pointing of the small hand of my watch to 7 and the arrival of the train are simultaneous events.”3

In this sentence Einstein only actually describes the fact that objects ( be they locomotives on tracks, or rotating hands on dials) can exist and move, and that their locations and motion can be compared.

He does not in any way point to a proof that there is a past or future, or 'time' or different 'times', in fact he seems only to show that objects  can exist and move, while also  assuming 'time' exists, and that a rotating hand on a numbered dial in some way shows times existence.

Importantly this detail gets more and more lost, as we seem to habitually conclude that Relativity's more complex conclusions back-track to a proof of time, whereas again they may only show that things can exist, move and  'change' - often at unexpected rates - but always only 'now' so to speak.

( See here >>∆ Relativity - The Electrodynamics of Moving Bodies paper.)  
∆ Relativity - The Electrodynamics of Moving Bodies paper.

That objects might just exist and change (IE never creating a past behind them or heading into a future)
 gives us a radically different and far simpler interpretation of our paradox.

Instead of assuming the billiard ball 'splits' in to two existences, one in the present and another in 'the past',

we consider that as the ball enters the worm hole it is just 'spaghettified'

And that this elongation becomes pronounced enough that the front of the ball can exit the wormhole, even though the rear is outside the hole.

in this way the FRONT of the ball is able to traverse the wormhole, such that it can exit, and hit its own REAR.

in this way the FRONT of the ball is free to impact, and affect the REAR, and deflect it in countless simple physical ways, but none that deflect the rest of the ball from its transit through the wormhole.

The idea that all parts of the ball exist and interact now, perhaps distorted in scale and rates of change, by relativistic effects, but not distorted over a thing called time,  complies with direct human observation ( we only ever see a constantly changing now) and eliminates the need for esoteric devices such as professor Hawkings's  "Chronology protection conjecture" ,

(conjecture -
noun: conjecture; plural noun: conjectures
  1. 1.
    an opinion or conclusion formed on the basis of incomplete information.)

Critically we can see how this solution allows even a living being , in this case a conveniently shaped snake, to likewise enter the wormhole, and interact with its own body.
Considering the possibility that Relativity might affect "rates of change" ( though not the flow of a thing called time) even allows us to consider how the snakes living head might interact with its dead tail... timelessly.

in this case the tail of the snake continues to expend the processes we call 'life', while the head is slowed down in these processes.

in this case the 'living' head could even meet, and devour it's own decomposing tail.
not because one is in the past, affecting the present, but because both are just in the (one and only ) present, changing at different rates.

we can note also here that we could pass two snakes through the tunnel in both directions and the effect would function both ways without confusion, let alone paradox. (and that drawing two snakes in a worm hole is hard to do neatley)

It is only the insistence that Relativity tells us about Space and Time, as opposed to Space and Change that makes the situation seem complex or impossible, but nowhere in Relativity does Einstein's work do more than assume that a changing device (such as a motorised hand on a numbered dial) counts as proof of a past and a future and time. 

(m.marsden, 'timelessness')