As with the game, it doesn't matter which particular fact you scrutinise first. Let us start with the Royal Opera House itself, the building rather than the opera company. It is being reconstructed at a cost of pounds 210m, of which lottery money covers pounds 78m. The balance is to be supplied by rich, private donors. It can be safely assumed that they won't put up the money if the building is turned into a receiving house for both the Royal Opera (with the Royal Ballet) and the English National Opera, currently operating out of the Coliseum.
This is because the implicit bargain with the donors - by subscribing you become the respected members of an exclusive, cultural club - would be broken. They would no longer feel that they were part of the opera company itself. Mr Smith appears to have come to realise this by indicating over the week-end that perhaps the English National Opera could go to the new Sadler's Wells theatre, so that the Royal Opera could continue on its own.
Another way of breaking the bargain - without which the building work will never be finished - would be to put the Royal Opera into liquidation. As a result, any donors, such as Lord Sainsbury and Vivien Duffield, who have topped up their funding with loans, would find that they would never recover what they have lent rather than given. Such treatment would likewise discourage further private funding. Whether the Royal Opera fails as a going concern will be decided tomorrow when Lord Chadlington, the chairman, presents his new plans to the Arts Council. The temptation to force the company into liquidation should be resisted.
It is because the management has been so spectacularly incompetent that a widespread desire to punish the board with bankruptcy exists. But the awkward fact to examine, exemplified in the person of Sir Jeremy Isaacs, who had a lengthy stint as general director until earlier this year, is that managerial ineptness has been combined with artistic excellence. By common consent, the recent work of the company has been as good as any done in the past 30 years and well up in the world league.
Sir Richard needs to propose, therefore, a new management structure for the companies that secures strong commercial ability as well as artistic flair and which does away with cronyism in the boardroom. In turn this means that so long as the companies are in receipt of public funds, appointments should require approval by the Arts Council or a government minister.
Let us next turn to the piece in the jigsaw puzzle marked "ballet" and at the same time pick up a second awkward shape - the Coliseum building. There is no logical reason why the Royal Ballet should be yoked with an opera company. Such pairing is relatively rare around the world. I suspect the Royal Ballet has always been treated as the junior partner and suffered as a result. As for the Coliseum, for some time the English National Opera has wanted to leave because the building has poor facilities backstage and is very dilapidated.
There is a lot to be said, therefore, for the Secretary of State's suggestion that the Coliseum should become a dance house, providing a home for an independent Royal Ballet, the English National Ballet and visiting companies. In the new circumstances, the Royal Ballet could apply to the National Lottery for funds to renovate the Coliseum. In turn the English National Opera would move into the new Sadler's Wells theatre, where it could retain its integrity as an independent company, with its own supporters and traditions.
As a result of these moves, rather than the present situation, in which two opera companies and a ballet company operate from two theatres, the three companies would each have a permanent venue. In particular, following the departure of the Royal Ballet, the Royal Opera would have extra nights at its disposal each week. Would such extra capacity help bring these expensive arts to ordinary people at affordable prices? There are two parts of this problem - the cost of seats and touring outside London.
High prices are mainly a feature of the Royal Opera; the Royal Ballet is cheaper and so is the English National Opera. A bold solution would be ruthlessly to employ the mechanism of the marketplace. The Royal Opera, on one of the evenings each week formally used by the Royal Ballet, could charge seat prices equivalent to what might be paid at the theatre or even at the cinema. It would revert to its much more expensive "normal" tariff during the rest of the week. Rich and poor opera fans alike would attempt to book for the "cheap" night, which would be quickly sold out.
Of the many disappointed customers, the rich, together with corporations who take guests to the opera as a form of high-class entertainment, would then apply for expensive seats during the rest of the week; while those of modest means would wait to try again for the next cheap night a week later. Perhaps there would have to be a ballot.
This is probably as far as Sir Richard could go in completing the jigsaw puzzle. The Secretary of State will have to do the rest and start with the question of touring. It is colossally expensive for the national opera and ballet companies to go round the country with their huge orchestras and elaborate sets. Nor would international stars necessarily sing or dance outside London. Mr Smith should accept this and try instead to strengthen our excellent regional opera companies.
There remains the most difficult problem of all - government subsidy at its present reduced level and box office receipts do not together provide sufficient funds to enable high quality performance to be achieved consistently. This is why Mr Smith has indicated that the Royal Opera House may have to be privatised - in other words stop getting any state help at all, so that additional subsidy could be made available to the other national companies.
Examples are given, such as the summer seasons of opera at Glyndebourne, which are financed entirely from ticket sales and private donations, or the Metropolitan Opera in New York, which achieves higher standards, bigger audiences and lower prices than the Royal Opera House, all with minute public subsidy.
But Glyndebourne does not employ international stars and it changes its limited repertoire cautiously. And in the United States all the arts benefit from the fact that donors earn tax relief on their gifts. The Met itself has the extra strength that the rich in New York, compared with London, are more numerous and even more wealthy. It would thus be an enormous gamble to reduce gradually to zero the pounds 15m annual subsidy that the Royal Opera House currently receives and rely upon private finance to make up the difference and more. But this is how the jigsaw puzzle can be completed.