Can you solve it? Maths games with bad drawings

Today’s puzzles begin with a low-fi version of Countdown: you roll five dice and using the basic arithmetical operations aim to get as close as possible to a target number.

Then the questions get a bit trickier, and a bit more interesting.

1. Five nice dice

A roll of five dice produces the numbers 1, 3, 4, 5 and 6.

i) By combining the numbers using +, –, x and ÷, what is the closest you can get to the target of 55?

ii) What is the largest whole number you can reach?

iii) What is the largest whole number you can reach using only subtraction? (tip: it’s a positive number.)

iv) What is the largest whole number you can reach using only division?

Clarification: By combining the numbers I mean creating an expression using addition, subtraction, multiplication and division (and as many brackets as you like.) For example, using the five numbers above, you could combine them in this way: ((5×6)+ 4) ÷ (3 – 1), which is equal to 17. You can use as many, or as few, plus, minus, times and division signs as you like. You must use all five numbers.

2. Five nicer dice

You roll five standard dice.

i) What is the probability the five numbers can be combined to reach 0?

ii) What’s the probability that the maximum result is obtained simply by multiplying them all together?

iii) What’s the probability that the maximum result is obtained uniquely by multiplying them all together?

iv) What is the probability that the five random dice cannot be combined to reach any of the targets from 33 to 99?

Today’s problems are adapted from Math Games with Bad Drawings, the new book by superstar maths communicator Ben Orlin. It is a beautifully produced compendium of maths games that is destined to be a classic of popular maths literature. Like his previous books it has charming sketches of stick figures all the way through.

The idea of the book is that maths can be playful – and social. Almost all the games he writes about are for two or more players, and are to be enjoyed by anyone aged 10 and up.

For example, the game from which today’s puzzles are adapted is as follows. One player calls out a number between 33 and 99. This is the target number. The player then rolls five dice. Now all players have two minutes to get as close to the target number. Your score is your distance from the target, and, evidently, lower scores are better.

I’ll be back at 5pm UK with the solutions. NO SPOILERS. Instead, please discuss your favourite maths-based games.

Thanks to Ben Orlin. Math Games with Bad Drawings is out now.

I set a puzzle here every two weeks on a Monday. I’m always on the look-out for great puzzles. If you would like to suggest one, email me.

I’m the author of several books of puzzles, most recently the Language Lover’s Puzzle Book. I also give school talks about maths and puzzles (online and in person). If your school is interested please get in touch.

On Thursday 21 April I’ll be giving a puzzles workshop for Guardian Masterclasses. You can sign up here.