Welcome Mathematicals to another story about Algebrus Mathema, the most famous mathematical detective in the world.

Whether it’s sequences or time, weights or algebra, Detective Mathema has the power to solve the puzzles.

Today’s story is about Sir Sidney Chirrup, a man who loved racing pigeons. He would get them to race other pigeons in contests the length and breadth of the land. He would get them to race each other. He even (truth be told) enjoyed racing them himself although he seldom won. His birds had a knack for a win. One day he came into some money and decided to buy as many pigeons as he could and visited the local pigeon farm to buy 20 new exquisite birds that had been imported from Africa.

“They are excellent and of such high pedigree. I will surely win all the races with these. But how much will they cost?” Sir Sidney asked.

“Ah, they’re not going cheap” said Alphonso, the bird breeder.

Sir Sidney delicately pointed out that they were in fact going cheep. Cheep being a noise common to most birds, pigeons no less than any other. Alphonso gave Sir Sidney Chirrup a look which suggested that this was not the first time such an observation had been made. He continued:

“These twenty prime pigeons will be one million pounds in all”.

“Oh dear, that’s much more than I expected” said Sir Sidney, somewhat sadly. Alphonso scratched his chin and a smile appeared on his face.

“Well, you could pay £1 for the first bird, £2 for the second, £4 for the third, £8 for the fourth and so on, the cost doubling for each bird.”

“Oh, that sounds much better value, I will pay you that way!”

But when Sir Sidney had paid the final bill, he was shocked at how much it had come to – £1,048,575 in total, and sought out Detective Mathema to explain what on earth had happened.

“Ah, it is simple” said Detective Mathema. “Rather than pay £1 million in one lump sum, you paid £1 for the first, doubling the rate each time, so £2 for the second, £4 for the third, £8 for the fourth and so one. The 20th bird cost you £524,288.

“But that doesn’t equal £1,048,575 or even close to £1 million. I’ve been robbed!”

“No, Sir Sidney, you forgot to add up the cost of each bird. £524,288 was simply the cost of the very last one. There were 19 others remember. The first costing £1, the second £2, then £4, then £8, and so on – in total, the 20 birds added up to £1,048,575”.

“Oh dear” replied Sir Sidney, thinking about the letter he could soon expect from the bank manager.

“If I wanted a big bill I’d be buying ducks not pigeons” – a joke he felt Alphonso might have liked.

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The Adventures of Detective Mathema with support from The Ernest Cook Trust.

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