When the world tunes into the Olympic Games it will think that, for the main part at least, it is watching the very best athletes in the world across a range of sports. To some extent that will be wholly true, the likes of Michael Phelps, Ryan Lochte, Kosuke Kitajima, Federica Pellegrini, Rebeccas Adlington and Soni, Stephanie Rice and many others of that ilk supreme examples of excellence in the pool.
The Olympic movement is one for all, a world in which universality plays a part, with lesser standards acceptable in the realms of representation: the "taking part" end of it all.
Then there's a third and complex sum in the equation: the rules that men make up as they go along and the variations in national selection policies, allowed on issues of excellence but not so when it comes to nations deciding that they don't want to include any in their number who have been proved to have cheated.
This is where the sums begin in swimming under the FINA criteria, in part governed by IOC guidelines:
FINA has sent out its invitations to nations that have qualified for relays and to those who made B cuts. There are between 50 and 100 swimmers around the world in each of the 13 solo events who would qualify on that basis. In fact, just 6 in each event will make it to London. There must be a lot of disappointment out there on consideration of a list that includes a few oddities, among them Stephanie Horner listed as Australian when she is in fact a Canadian 400IMer.
In the mix is the case of Britain's Siobhan-Marie O'Connor, listed as having had her invitation to compete at London 2012 accepted by the British Swimming federation for a B cut time of 1:09.15 in the 100m breaststroke. O'Connor clocked 1:08 flat at last-chance saloon trials in Sheffield last week. Inside FINA A cut she is on the home Games team anyway. So does her place go to someone else now?
If her case is quirky, more troubling is that of 16-year-old would-be Britain teammate Molly Renshaw. At Olympic trials proper, a 2:26.63 won the British title and automatic London 2012 selection for Stacey Tadd. Renshaw was inside A cut too, just 0.18sec away. An oddity of British selection policy was that the second-placed swimmer hads to go faster than the winner. What? I hear you cry. True - if you finished second, you had to do a time equivalent to world top 16 on the SwimNews World Rankings for 2011, that time a touch faster than FINA A cut, the standard required by all winners.
Renshaw raced again at second-chance saloon but missed the mark by a small margin. She appealed the decision not to select her and under the letter of the law lost her case, the policy, regardless of arguments for or against, clear.
In a statement today, the federation said: "British Swimming can confirm the appeal by Molly Renshaw against non-selection for the 2012 Olympic Games has been dismissed today by an independent appeals panel in London. British Swimming Chief Executive David Sparkes said: "Molly has been swimming incredibly well this year but, while she has achieved a great deal, she didn't meet the qualifying standard for the Olympic Games. Molly will now focus on the European Junior Championships and US Open and we feel confident she will give her best there and come back with some success. We fully understand why Molly challenged the selection policy but again the result shows the selection panel has been consistent and robust in following the policy."
British Swimming will work with Molly now to ensure she achieves her goals for the rest of the season, and prepares for next year when she will be a strong contender for the World Championship team before looking towards the Commonwealth Games in 2014.
Renshaw can expect much support within a system that has improved imeasurably in the hands of Bill Sweetenham, his successors Denis Pursley and Michael Scott and a world-class team of largely home-grown coaches.
But she may still feel slighted. Take the list of B cut invitations from FINA for the 200m breaststroke: from a 2:26.91 for Norway's Sara Nordenstam through swimmers from Iceland, Ukraine, Slovenia and Turkey to a 2:29.27 for Martina Moravcikova of the Czech Republic. Renshaw will watch them racing at the Olympics and could be forgiven for feeling a touch sorry for herself. Hopefully she will do what Rebecca Adlington and Keri-Anne Payne (both invited by Sweetenham to the Athens pre-Games camp to keep Rebecca Cooke company and gain experience) did back in 2004 when they missed the cut for the Olympics in Athens but fought on in pursuit of a better day, one that for them went all the way to the Olympic podium.
Sport, of course, is full of the stuff off men making up the rules as they go along. During the finals session this evening in Omaha, for example, there was a fine ceremony to honour all those pre 1984-Olympic relay swimmers who helped teammates make a final but did not race themselves in a final and therefore received no medal.
Since 1984, all relay members, up to six, from heats and finals, get a medal if the fastest quartet make the podium in the final. Before the Los Angeles Games, only those who actually raced for the medal in the final got a medal.
There are merits in both ways of doing things, many a worthy heats effort having helped teammates a touch faster conserve their energy in readiness for racing into the medals; and a fair few who swam well shy of what it would have taken to make a podium but able to trade on a status of "Olympic medallist" for the rest of their lives.
There is a difference between swimmer A who makes the podium off their own steam and swimmer B being granted a medal even though they swam at a speed outside what it would take to make the world's top 30 in a solo over the distance they covered. Even so, a good thing to recognise those who made a fine contribution to the success of others through their own hard work and commitment.
Less comfortable is the notion that world-class swimmers will sit at home watching the Olympic Games this summer while others slower than them will race in realms that will forever allow them to say "I was an Olympic swimmer". The assumption is that those who swam at the Olympic Games were the best there were at the time (plus a few there to "take part" under universality principles). That is clearly not the case.
Rather than the Games representing four heats of the best 30 in the world two per nation, plus a couple more for universality, the numbers at the Olympic Games look more like this on world ranking (an example taken from the women's 200m breaststroke):