## 13 Jan 2013

### Only Three Dice

It's raining outside, and Alice and Bill have to stay indoors. They are looking through Alice's box of toys when she exclaims,"Found it! Let's play Number Quest!" "Sounds like a maths game." says Bill, unenthusiastically. "Yeah, but it's dead easy to play," gushes Alice, "Even for you!" She gave Bill a mocking look that meant he had no choice in the matter. Alice beamed.

They unpacked the game. It was, indeed, a rather simple game; roll the four dice and use any of the four arithmetical operations (and brackets if needed) to make the highest possible number between 1 and 100. The higher the number the more prize money you win and you place your counter on that number so nobody can use it again. But as the game progresses it starts to get harder as many of the top numbers are covered with tokens.

"Oh no!!" shrieked Alice. "I can't find all four dice! I can only find three." Bill thought something really dramatic had just happened, like a spider crawling out of the box. "Maybe we could play something else." he said, trying to put his positive voice to good use. But Alice didn't even reply; she was too busy taking all the games out of her games box to look for the missing die. After a good 10 minutes of rummaging, she admitted defeat.

"Let's play with three dice!!" Alice exclaimed. "Can we still make all 100 numbers, though?" asked Bill. "Probably not," Alice said thoughtfully, "But let's find out!" Bill was just not going to get out of playing Number Quest.

The Question

What are the two smallest numbers that cannot be made with just three 6-sided dice using just arithmetic operations?

Also, what are the two largest numbers that cannot be made?

This is not a Prize question, so feel free to comment.

Good Luck!

Maths Questions posted on Mondays, Wednesdays and Saturdays.
Prize Maths Questions (PMQs) posted on Fridays.

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If you're feeling tired, defocus your eyes on this stereoscopic photograph.

From Wikimedia [public domain]
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