15 Mar 2013

Prize Maths Quiz: A Bag of Odd Numbers (PMQ10)

"I want to show you something.
I can't figure out this week's PMQ."
Louis Mordell and Gabo Szego

Alice was getting ready to impress her smart cousin Zeta. She went downstairs and could see Zeta sitting in the garden immersed in a book.

“Cousin Zeta, I have a little puzzle for you.”

Zeta barely looked up,”A bag of tricks, no less!”

Alice started to fidget with her small bag and the counters rattled her nerves.

“In this bag are all the odd numbers...”

All of them?” interjected Zeta.

“Well, what I meant...” if you’d let me finish, thought Alice,”... was that there are all the odd numbers in order, 1, 3, 5 and so on. Now, I’m going to pick one of them at random and I’m just going to tell you what the average of the remaining counters is. Your mission is to figure out which number I took out.” Alice beamed; she’d been rehearsing this for the last half an hour.

“How many counters are there in total?” Zeta placed her book down, intrigued by her little cousin.

“Can’t tell you that!”

“So, you have a bag full of odd numbers. I don’t know how many, but they are consecutive numbers from 1 upwards. You take out one of the numbers, and again I have no idea what it is. The only thing you’re going to tell me is the mean of all the remaining numbers in your bag. Is that it?”

“Yes! Perfect!” beamed Alice. “Shall we play?”

“Go on then! But, let me pick the number.” Zeta smirked.

Alice thought about that for a few seconds. “OK, but you can’t look at it! Just pick it with your eyes closed and give it to me.”

“OK, let’s play!” Zeta shut her eyes so tightly that she made an ugly grimace. She picked a number from the bag and gave it to Alice. “Can I open my eyes now?”

“Yes, yes!” Alice had turned around and was busy scribbling a few calculations in a small pocket book. “OK, done! The average of all the numbers left in the bag is ... 51.2! Now you have to tell me which number you picked.”

Zeta seemed to blank out and stared at a picture in her mind.

“I have a pen and paper if you need them.” said Alice. She was just happy that Zeta didn’t just come out with the answer in seconds.

“I have a brain.” Was Zeta’s laconic reply.

“OK, I’m going to get a drink – it’s very hot out here. Would you like one?”

Zeta didn’t reply.


Can you find the number that Zeta pulled out from the bag of consecutive odd numbers?






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