Yes, at long last, here it is, the legendary (!?) missing...
SOPHIST's CHOICE Why 1=0 and how this proves Barrow should not have lost their Football League place in 1972
Way back in issue 008 (Nov 91), when we first began our series of Sophist's Choice, we promised that we would establish, once and for all, that Barrow should not have been voted out of the Football League in 1972. We weren't going to do this by just rehashing the usual arguments about Barrow not being bottom of the table, or by pointing out that the League breached its own rules when it held a new ballot after the initial vote was tied with Hereford United. No, though these points remain valid of course, we were going to take a fresh approach and demonstrate our case through the scientific application of the ineluctable logic of pure mathematics. This promise, however, was contingent on our former Editor either remembering or being able to work out an algebraic proof that 1=0 that he had first been shown as a mere whippersnapper back in Mr Punton's second year Maths class at the old Grammar School in 1970. Typically, in the end he did neither and got someone else to calculate it for him. But now that he has, we can at last substantiate, conclusively and beyond any shadow of a doubt, the previously preposterous proposition that one equals zero, and zero equals one. "Whaaat!? They've lost it this time!" I hear some of you saying, reaching for the back button already. Well, I shouldn't do that if I were you. For Old Ed also informs us that Mr Punton, in his other capacity as Physics master, informed his class on more than one occasion that there was going to be a total eclipse of the sun in Cornwall in August 1999. So as Old Ed quite rightly suggests, if his old tutor predicted that correctly, then why shouldn't he be right about one equals zero too? Ah, you're back. So, enough time wasted, here now is that arcane and esoteric formula as it was chalked on that blackboard so many years ago...

Let:	a =	b
Multiply both sides of the equation by a:	a2 =	ab
Add (a2 - ab) to both sides:	a2 + a2 - 2ab =	ab + a2 - 2ab
Factorise and simplify:	2(a2 - ab) =	a2 - ab
Divide both sides by (a2 - ab):	2 =	1
Subtract 1 from each side:	1 =	0

Now we have established that one is indeed equal to zero and vice versa, then it follows that the one point awarded for a draw in football is also zero. Equally, the reward for a defeat is now one point instead of that big fat round nothing so often recorded before. Indeed, since we also proved that one equals two, we could argue that only one point should be awarded for a win and that even that point should be deducted, since one is also equal to zero. But we'll leave that aside as things are going to get complicated enough around here without bringing that into it! But by now the implications for Barrow's predicament back in 1972 should be becoming clearer. In those days, of course, the bottom four teams in Division Four all had to apply to the League for re-election. Thus, the end of each season saw a mad scramble to finish at least 20th out of 24. In 1972 Barrow finished 22nd, two places shy of that crucial position, which was occupied by Chester City (see figure i) below). So if we are to prove that Barrow shouldn't have had to seek re-election then we must establish that they actually obtained more points than Chester. Now, bearing in mind the proof we determined above, then if one equals zero, then it must be valid to count Barrow's defeats as one point, and the Chester draws ("Oooh Arthur, where's me washboard?") as no points. In this case, while Chester would record only 20 points, Barrow finish with 59, well clear of the drop zone. Okay, no doubt you'll have spotted our somewhat dishonest logical flaw in that one, but the fact remains that if both clubs' draws are counted as no points, and both clubs' defeats as one point, then in any of the three possible permutations that allows (see figures ii) - iv) below), Barrow finish above Chester every time. In fact Barrow actually finish in nineteenth position, two places above the bottom four, as these corrected tables show for each calculation. The numbers in brackets are the historically accepted positions and points, now definitively proven here to be wildly, and perhaps wilfully, inaccurate.

Fig i) - FOOTBALL LEAGUE DIVISION FOUR 1971-72 - Bottom eight
	P	W	D	L	F	A	Pts
17 Aldershot	46	9	22	15	48	54	40
18 Hartlepool United	46	17	6	23	58	69	40
19 Darlington	46	14	11	21	64	82	39
20 Chester City	46	10	18	18	47	56	38
21 Northampton	46	12	13	21	66	79	37
22 Barrow	46	13	11	22	40	71	37
23 Stockport County	46	9	14	23	55	87	32
24 Crewe Alexandra	46	10	9	27	43	69	29

Fig ii) - FOOTBALL LEAGUE DIVISION FOUR 1971-72 Revised bottom eight based on one point for a draw and one point for a defeat
	P	W	D	L	F	A	Pts
17 Exeter City (15th - 43pts.)	46	16	11	19	61	68	62
18 Darlington (19 - 39)	46	14	11	21	64	82	60
19 Barrow (22 - 37)	46	13	11	22	40	71	59
20 Northampton (21 - 37)	46	12	13	21	66	79	58
21 Chester City (20 - 38)	46	10	18	18	47	56	56
22 Crewe Alexandra (24 - 29)	46	10	9	27	43	69	56
23 Aldershot (17 - 40)	46	9	22	15	48	54	55
24 Stockport County (23 - 32)	46	9	14	23	55	87	55

Fig iii) - FOOTBALL LEAGUE DIVISION FOUR 1971-72 Revised bottom eight (no points for a draw and no points for a defeat)
	P	W	D	L	F	A	Pts
17 Exeter City (15 - 43)	46	16	11	19	61	68	32
18 Darlington (19 - 39)	46	14	11	21	64	82	28
19 Barrow (22 - 37)	46	13	11	22	40	71	26
20 Northampton (21 - 37)	46	12	13	21	66	79	24
21 Chester City (20 - 38)	46	10	18	18	47	56	20
22 Crewe Alexandra (24 - 29)	46	10	9	27	43	69	20
23 Aldershot (17 - 40)	46	9	22	15	48	54	18
24 Stockport County (23 - 32)	46	9	14	23	55	87	18

Fig iv) - FOOTBALL LEAGUE DIVISION FOUR 1971-72 Revised bottom eight (no points for a draw and one point for a defeat)
	P	W	D	L	F	A	Pts
17 Exeter City (15 - 43)	46	16	11	19	61	68	51
18 Darlington (19 - 39)	46	14	11	21	64	82	49
19 Barrow (22 - 37)	46	13	11	22	40	71	48
20 Crewe Alexandra (24 - 29)	46	10	9	27	43	69	47
21 Northampton (21 - 37)	46	12	13	21	66	79	45
22 Stockport County (23 - 32)	46	9	14	23	55	87	41
23 Chester City (20 - 38)	46	10	18	18	47	56	38
24 Aldershot (17 - 40)	46	9	22	15	48	54	33

So there you have it. QED. We can state now, quite unequivocally, that Barrow should never have even had to apply for re-election in 1972 in the first place. At last, over a quarter of a century of injustice has been exposed. We demand to be re-instated to our rightful place in the League. Now! We don't care who they chuck out to make way for us (though it would be nice if it were Carlisle). You can start making your travel plans to Torquay and Leyton Orient now, fellas! PS If three points for a win had been in operation in those days, then we wouldn't have to resort to dodgy algebra to keep Barrow in the League. Under the current system, Barrow would again have finished in nineteenth position, this time with 50 points, ahead of Aldershot (49 pts.), Northampton (49), Chester (48), Stockport (41) and Crewe (39). PPS By the way, we're sending this proof to the Conference. We figure they've probably run out of dodges to keep Welling United in the league and might need it if they're going to rescue them again this season.
Special thanks to Gary Savage for working out the proof 1=0. Issue 044 - March 2000
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