Puzzle 5

Following the failure of her popular yet unprofitable apple shop business, Hazel decides to turn her attention to citrus fruits. Having recently developed an interest in the number \(x\approx 1.47\) which satisfies \(x^3=x^2+1\), Hazel decides to introduce her own currency, with coins of value 1 pound, \(x\) pounds and \(x^2\) pounds. She further decides to price each piece of fruit as a power of \(x\).

The most expensive item of fruit in the shop, the blood orange, is priced at \(x^{20}\) pounds. How many coins must change hands for a customer to buy a blood orange?