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ITV is risking censure and a possible big fine unless it can explain the answer to a puzzle on a late-night premium-rate phone quiz show that no one can solve.

The riddle, which would breach broadcasting guidelines if deemed unreasonable, is the latest blow to an industry battered by investigations into alleged interactive rip-offs.

The Times challenged great thinkers and academics but none could crack a puzzle intended for an early-hours viewing audience of drunks and poor sleepers.

Mathematics professors, the high-IQ organisation Mensa and the lateral-thinking pioneer Edward de Bono all failed to come up with a solution. Finally the cryptanalysts of Bletchley Park, which hastened the end of the Second World War by cracking the Nazis’ secret ciphers, provided an answer. But ITV said that the codebreakers were wrong.

The puzzle was broadcast on the Make Your Play programme, which begins broadcasting on ITV1 at midnight.

The on-screen question, headed Add the pence, listed: Two pounds, 25p, £1.47, 16p, Fifty pence. A prize of £30,000 was offered. The phone tariff to enter was 75p. The presenter told viewers: “Takes a bit of working out, I know, but it’s worth it: Thirty grand!”

Three and a half hours into the programme, the host simply announced that the answer was 506 — and that no one had won.

A reader of The Times said: “I was prepared to be shown how it was done. Then, to my complete disbelief, he moved straight to another game without explaining how you actually got 506.”

An ITV source said the company refrained from explaining puzzle answers because “we feel it would ruin the game for some people”. The puzzle was checked by the office of ITV Play’s controller, William van Rest, who is a maths graduate, before it was broadcast.

So The Times asked some of the world’s brainiest people to say where 506 came from.

Marcus du Sautoy, Professor of Mathematics at Oxford University, showed the puzzle to fellow professors at a New Zealand university. “No one can find a good reason for the answer being 506.

“That’s not to say that we haven’t missed some clever perspective but there was a growing sense amongst those I talked to that this could well be a scam,” Professor du Sautoy said. “Anyone adding up the pence in the obvious way gets to 438. So there is a missing 68 pence. There might be tricks like you have to include the pence in the question ‘Add the pence’ but I’d be intrigued to see what the rationale is behind the solution of 506.”

Edward de Bono, author of the forthcoming How To Have Stunning Ideas, was sent the question. His spokeswoman said: “The only thing he can think of is that they are possibly mixing old pence with new. He hasn’t gone any further into it but it’s an idea.” British Mensa sent the puzzle to several members but, a week later, none has produced an answer.

Eventually, the Bletchley Park Trust found an expert who reached a solution after writing a code-cracking computer program. The Times presented the answer to ITV but a spokeswoman said: “It’s nowhere near as complicated.”

The regulator Ofcom said that it was investigating 20 complaints against Make Your Play. Ofcom has already reprimanded ITV Play for selecting “Rawlplugs” as the answer to a contest to name the contents of a woman’s handbag. Repeated breaches of broadcasting rules can lead to fines of up to £250,000 and suspension of licence.

John Whittingdale, the chairman of the Commons Culture Committee, said: “The idea that you could have an answer that can’t be arrived at is unacceptable.” Virgin Radio is the first radio station to join the phone-in controversy after it broadcast appeals for record requests on a prerecorded show. Suggs, who was hosting an apparently live show, was on Ant & Dec’s Saturday Night Takeaway on ITV1 at the time.

Virgin admitted that an edition of the show broadcast on October 21 last year had been prerecorded.

Genius who gave up and went to bed

The only expert who answered the puzzle was Frank Carter, a retired maths lecturer so fiendishly clever that he was chosen to help to rebuild a machine that cracked the Nazis’ secrets.

He was given the puzzle and ITV’s stated answer of 506 and asked to get from one to the other. A former schoolmaster, he was withering about the calibre of the question.

“There would have been a range of answers that you could conceivably come up with which would be equally valid and it is a matter of luck whether you get it right,” he said. “I don’t think it is a good question because it is open ended and you could argue quite lucidly for a number of different answers. It is like having a mathematics question in a school examination in which you have half a dozen answers all equally plausible.”

Mr Carter spent 90 minutes pondering the question before he “got fed up, gave up and went to bed”. The next morning he wrote a computer program using the childish code A for 1, B for 2 etc. Within half an hour he put together a form of letters that could be shoehorned into a potential answer. ITV says that his explanation was wrong.

The elite calibre of mind required to reach any solution to ITV’s question is highlighted by Mr Carter’s reputation. He is one of a team recreating the Turing Bombe, the celebrated machine that cracked the Nazis’ Enigma code and hastened Hitler’s defeat by two years.

His working replica can be seen at Bletchley Park museum near Milton Keynes.

dkennedy@thetimes.co.uk

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