"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?

**You can receive these questions directly to your email box or read them in an RSS reader. Subscribe using the links on the right.**

**How to Enter****Send your complete solution by email to pmq10descartes@giftedmaths.com.**This email address shall be removed after the competition closes to avoid spam.

**This PMQ10 competition closes on MONDAY 18 March at 23:59 GMT - one extra weekday from now on**.

**The Prize****The prizes for this PMQ are 3 free places in our Online Maths Club for 2 months.**The very first correct solution will receive a prize plus two others randomly selected from all the other correct answers. The email time stamp shall determine the order of entries received.

**Quick Rules****Look at the expanded rules on our PMQ page.**

Anybody can enter our Prize Maths Quiz; adults and students.

Use the official server clock in the right column to avoid late entries.

All emails and email addresses sent to us will be deleted after the winners have been processed.

DO NOT submit your entries to the comments section at the bottom of this post.

You CAN discuss it there but you must email us to enter the competition.

We adhere to COPPA guidelines regarding children's online safety and security.

Enjoy the challenge!

**Send us your solution! You can't win if you don't participate.**

Then tell your friends!