The sequence 2, 3, 5, 6, 7, 10, 11, ... consists of all positive integers that are not a square or a cube. Find the 500th term.

My boys learnt to count in a base different from 10, so that instead of writing 107, they write 255. In what base did they learn to count?

Mr Morgan G Bloomgarten, the millionaire, known in the States as the Clam King, had, for his sins, more money than he knew what to do with. It bored him. So he determined to persecute some of his poor but happy friends with it. They had never done him any harm, but he resolved to inoculate them with the "source of all evil". He therefore proposed to distribute a million dollars among them and watch them go rapidly to the bad. But he as a man of strange fancies and superstitions, and it was an inviolable rule with him never to make a gift that was not either one dollar or some power of seven - such as 7, 49, 343, 2401, which numbers of dollars are produced by simply multiplying sevens together. Another rule of his was that he would never give more than six persons exactly the same sum. Now, how was he to distribute the 1,000,000 dollars? You may distribute the money among as many people as you like, under the conditions given.

Three countrymen met at a cattle market. "Look here, " said Hodge to Jakes, "I'll give you six of my pigs for one of your horses, and then you'll have twice as many animals here as I've got." "If that's your way of doing business," said Durrant to Hodge, "I'll give you fourteen of my sheep for a horse, and then you'll have three times as many animals as I." "Well, I'll go better than that, " said Jakes to Durrant; "I'll give you four cows for a horse, and then you'll have six times as many animals as I've got here."

No doubt this was a very primitive way of bartering animals, but it is an interesting little puzzle to discover just how many animals Jakes, Hodge and Durrant must have taken to the cattle market.

P is a point inside the triangle ABC. Lines are drawn through P parallel to the sides of the triangle. The areas of the three resulting triangles with a vertex at P have areas 4, 9 and 49. What is the area of ABC?