Imagine a jigsaw puzzle of, say, 10 million pieces (or equivalently, 10 identical jigsaw puzzles of a million pieces each). Each piece is a perfect square, 1 cm x 1 cm, and all the pieces are completely blank, except that at the back of each piece there is a serial number. Imagine that this entire puzzle is to be contained in a frame 50 m long and 20 m wide. Imagine that a man starts to work on this puzzle when he is still a teenager, carefully selecting each piece before he fits it into place (since all the pieces are identical, the serial numbers are irrelevant - he makes his 'choice' of which piece to fit independent of them). Say it takes him a minute to decide which piece to fit next. Working an average of 10 hours a day - corresponding to a 12 hour work day with breaks for coffee, lunch, etc. (but taking no time off for weekends or holidays), it would take the man almost 46 years to complete this puzzle. At the end of that time, he would have created an original work, an arrangement of the 10 million pieces in an order that, on a purely statistical basis, would be unlikely to have ever been or be replicated by anyone else, even if there were, say, 6 billion other people working on a similar jigsaw puzzle. This would be his masterwork, the one thing that was truly and inalienably his own, the one thing that made him unique.
And yet, to an observer, it would be utterly indistinguishable from the 6 billion other such jigsaws being assembled all around him.
Next, imagine a game of scrabble where all the tiles are blanks. What word would you choose to place on the board first, assuming you were the first to move. What word do you think your partner / opponent would choose? How would you respond to him or her? What words would you never use? What words would you have to?
And when the board had all filled up with these imaginary letters, and no further words were possible, how would you score the game, given that blanks are worth nothing? Who would you say had won?
Imagine that we return to the jigsaw puzzle. Except this time the pieces (still blank, still numbered on the reverse) come in three different shapes, and can be combined only in certain ways. Moreover, there is no frame now, just a flat, limitless surface on which these pieces may be arranged.
You will say that this is no longer a jigsaw puzzle, but a game of building blocks, because the puzzle can now take on (within the limits imposed by the symmetry of the pieces) any shape the man chooses. This may be true, but the man does not see it that way. He is not looking to create a shape of his own, he is trying to discover the right shape that is hidden in the puzzle and that he knows exists though he does not have a picture of it (the game is borrowed from a friend, who has thrown away the box it came in and which contained, it is rumoured, an illustration of what the completed game should look like). This is the challenge the game has set him. That is the challenge he has set himself.
How is the man to proceed now? Suppose he lays out the game (taking, once again, a little under 46 years) and he gets the shape wrong. How would he know? Would someone tell him? And even if they did, could he correct the game once he found out? Could it be that just rearranging a few of the pieces would set it right? Or would he have to start all over again? (And would he have the time to do this? And how would he know how to get it right the second time around?)
Most importantly, though, where should he start?
Imagine a game of chess. Imagine that all of the pieces move the way they do in a standard chess game, except that it is not possible for any piece to retreat - a piece can move forward or sideways (forward implying, of course, towards the opponent's side) but not back. The only way that a piece can go back in the direction it has come from is to switch allegiance - go over to the enemy. An infinite number of such acts of treachery are allowed, however. Each piece can prove turncoat as often as it chooses, and at any point in the game - its change of side is to be negotiated between the two players any way they choose.
Is there a way to win this game (assuming that both players are playing to win)?
Next, assume that we add a further twist to the game described above. Assume that the rules of combat are reversed - that the objective is to not land in a square that is already occupied by an opponent's piece. This is a game of avoidance. Any piece that lands in a piece that is already occupied is removed from the board.
Is there a way to win this game?
Finally, back to the jigsaw puzzle again. A smaller puzzle this time, say only 100,000 pieces. All perfect squares, 1cm x 1 cm, a frame of 5 metres x 2 metres being provided. Only this time the pieces are not blank, but every piece is a very marginally different shade of blue. The pieces are not numbered or marked in any way, and the player can tell the difference between each piece only through visual inspection.
Obviously, it is now possible (at least for a careful observer) to tell all the different arrangements of the puzzle apart - because each will have its own distinct colour scheme. How is the man to go about his task now? Should he strive to discover a pattern in the colouring or create one? Should he try to match shade to shade, so as to produce an even gradation of colour (and if so, in what direction) or should he seek to dazzle by contrast? Should he use the fact that he now has only 1/100th as many pieces as he had earlier to spend more time making the decision on each piece (potentially spending as much as an hour and forty minutes on each one)? Or should he try to stick to his normal pace, and give himself the option of starting over (up to a maximum of a 100 times) if the puzzle doesn't come out right the first time?
Say he does the latter. How is he to compare one arrangement to another? What criteria is he to use? How is he to remember his prior arrangements well enough to be even able to compare?
And suppose that he does manage to do 99 of these arrangements, and decide, by some means which one of them was the best one. Unless it happens to be the 99th arrangement, how is he to reconstruct, exactly, the one that he preferred?