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Twenty Divided Into Six

Stage: 2 Challenge Level: Challenge Level:2 Challenge Level:2

A special welcome to the some newcomers, pupils at St Aldhelm's , C.E.V.A. Combined School in Poole in Dorset. We hope you continue to use the site and use all the great skills you showed this month to solve the NRICH problems. Thank you to special friends of NRICH - Moorfield Junior School , Tattingstone Primary School and Burgoyne Maths Club as well as Hutt Intermediate School in Wellington, New Zealand, Nan Hua Primary School , Singapore, Higher Bebington Junior School , Rosebank Primary School Leeds, and Lazonby C of E School.

Not only were there many responses there were also many solutions to this problem. Below, are a selection of the possibilities. I wonder if you could add to the list. In fact, I wonder if you could find a way of discovering how many possible answers there are to this question. Now there's a challenge!!

The problem took more planning and moving of numbers than people realised at first. One strategy was explained by pupils Emma and Hollie of Moorfields Junior School:

We added up all the numbers up to 20 and divided the total by 6.

Next step was described by their classmates Steve , Matthew and Luke:

... we paired up all of the numbers and then we started to move the numbers into groups.

Burgoyne Maths Club wrote:

After various attempts to make six equal piles with twenty cards, we abandoned this because we couldn't make twenty (the number of cards) with 6 different numbers.

Georgereported:

I began with 20+15 and....the rest was just matching up.

Have a look at these solutions to see if yours is amongst them or if you have the same solution. Perhaps you had some solutions from one set and some from another. Would it be possible to have one answer from many different sets of possibilities? What reason do you have for your answer to that question?

20, 15
1, 2, 3, 4, 5, 9, 11
18, 17
10, 13, 12
19, 16
6, 7, 8, 14
20,15
1, 14, 11, 9
10, 19, 6
3, 4, 12, 16
17, 13, 5
2, 7, 8, 18
20, 10, 5
15, 1, 19
6, 13, 16
3, 12, 2, 18
8, 7, 9, 11
17, 4, 14
20, 15
19, 9, 7
18, 17
16, 13, 5, 1
14, 12, 6, 3
11, 10, 8, 4, 2
17, 18
19, 16
20, 10, 5
4, 7, 15, 6, 3
12, 9, 14
13, 11, 2, 1, 8
17, 13, 5
9, 15, 11
1, 4, 12, 18
6, 8, 7, 14
19, 16
2, 10, 20, 3

These solutions were a selection representing the hundreds that were sent in. The senders of these were George and Matthew from Rosebank Primary in Leeds, Ong of Nan Hua Primary in Singapore, Martin from Robert Kett Junior School in Wymondham, Suzanne and Lucy from Lazonby C of E School, Joshua from Higher Bebington Juniors, Steve , Matthew and Luke of Moorfiled Junior School, and members of the Burgoyne Maths Club.

Towards the end of 2015 we had  solutions sent in from Ms. S. Fox's class at Cutteslowe Primary School in Oxfordshire. Two of them wrote letters to explain what happened and the class as a whole sent in their own solutions. Thank you for these more recent solutions. 

We also had this as a solution from year 5 at Southville Primary School, Bristol, UK:

We found out that it was impossible to make 6 unequal piles with 20 cards. Bailey said "It would work if we had 21 cards!"

We were able to make 35 with 6 piles, but some of them were equal.

Lucia made this combination:
1, 5, 12, 17
2, 8, 14, 11
3, 9, 19, 4
7, 10, 18
6, 16, 13
15 and 20

Thank you for your tricky problem! We enjoyed proving the first part was impossible!

Thanks for these words and solution