The London mayor, Boris Johnson, has proven that a high IQ and an Eton education are of little help when someone asks the cost of a cash tube fare from Angel to London Bridge live on air.

After struggling for a minute to answer the question on an LBC radio phone-in, Johnson also failed two questions from a quiz presented like an IQ test and refused to attempt to answer a third, saying: "No one said IQ is the only measure of ability".

It was his first public appearance since he caused controversy by suggesting some people struggle to get on in life because of their low IQs, adding that the bigger cornflakes tended to end up at the top of the packet.

Struggling himself in the course of the 45-minute phone-in to avoid tumbling ever further down the packet, Johnson also claimed that Marble Arch tube station had once been called Selfridges, a claim he has made twice before but is untrue. There once had been a plan to rename Bond Street but it never came to pass.

He also revealed that in his six years as London mayor he has never met the leader of the RMT union, Bob Crow, even though the union leader represents London's bus and tube workers.

In the first IQ question he was asked: "A man builds a house with four sides of rectangular construction each side with southern exposure. A big bear comes along. What is the colour of the bear?"

Johnson said the bear was probably brown, before admitting that he did not have a clue. The answer is white because it must be a polar bear on the north pole.

Next he was asked: "Take two apples from three apples and what do you have?" Johnson said: "Loads of apples." He then changed his answer to one apple. The answer the questioner wanted was two apples.

He was then asked: "I went to bed at eight in the evening last night and I wound up my clock and set my alarm to sound for nine o'clock in the morning. How many hours sleep did I get?" But Johnson refused even to attempt an answer.

"No one said IQ is the only measure of ability," he said, perhaps sensing that he was nestling very close to the bottom of the cornflake packet by then.

Johnson sought to defend his speech last week that was interpreted by some as saying greed is a good motivator and needs to be encouraged.

"There is too much inequality," he said. "My speech was actually a warning against letting inequality go unchecked." He then said that "in last 20 years there has been a widening of income between rich and the poor".

He added: "What hacks me off is people with ability have found it very difficult to progress in the last 20 years. The key thing I said is inequality is only tolerable in our society if you look after those who are finding it tough to compete and where people have ability they are allowed to get on."

He turned on some of his critics, including the deputy prime minister, Nick Clegg, saying "people are entitled to wilfully misconstrue what I said if they so choose".

Labour has fiercely attacked Johnson over the issue, with the shadow education secretary, Tristram Hunt, challenging Michael Gove to denounce the "unpleasant whiff of eugenics" contained in the comments.

In what seemed to be the longest minute ever broadcast on radio, Johnson struggled to work out the cost of a one-way cash fare from Angel, his local station, to London Bridge, his place of work. He mumbled and hummed before pronouncing £6.70 and then seemed not to know the basics of the Underground zoning system.

He was then asked about the talent show The X Factor, saying he did not have the foggiest idea about it, and whether he knew who was in the I'm A Celebrity Get Me Out of Here jungle, answering: Nick Clegg, Vince Cable and Ann Widdecombe.

"I have not the faintest idea and why should I?" he added.

Johnson revealed he was considering sponsorship of tube stations, saying if Samsung wanted to change Tottenham Court Road to the company name, he was not against the idea in principle so long as the right fee was paid. The cost of changing names on the maps and leaflets would be £4m alone.

Johnson, who was speaking on the day of the latest tube fare rises, said he was looking at ideas such as ticket offices being used as coffee shops and retail spaces.

He said: "It will be much better to get people out from behind plate glass where they are serving fewer and fewer tickets and to get them on to platforms and concourses."