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This challenge obviously led to some interesting explorations! Here's a solution sent in by Maicy, Caitlan and Taryn
 from Saxmundham Primary School in England


Calculation Method Answer
3 x 6 mental 18
18 - 3 mental 15
15 ÷ 3 mental 5
5 x 2 mental 10
10 x 2 mental 20
20 + 1 mental 21
21 ÷ 3 mental 7
7 x 2 mental 14
14 - 2 mental 12
12 ÷ 3 mental 4
4 x 2 mental 8
8 - 5 mental 3

if you use this card from another set $10 \times 3$ the card that follows is $30$.

if you use this card from another set $12 \div 2$ the card what follows is $6$

Myles, Joshua and William
 also from  Saxmundham Primary
 School from England
 also sent in;
 
Sum Method Answer
7 x 2 mental 14
14 - 2 mental 12
12 ÷ 3 mental 4
4 x 2 mental 8
8 - 5 mental 3
3 x 6 mental 18
18 - 3 mental 15
15 ÷ 3 mental 5
5 x 2 mental 10
10 x 2 mental 20
20 + 1 mental 21
21 ÷ 3 mental 7

If we used the card from a different set $10 \times 3$ it would make $30$.
If we used $12 ÷ 2$ it would make $6$.

Ashton from Raynsford Voluntary Controlled First School in England sent in this solution;

I started with $8 - 5$ at that $3$ then I found the problem starts with $3$
and that was $3 \times 6$ then I work it out and then I did what you have just done
again so it will be $18 - 3$ then $15 ÷ 3$ then $5 \times 2$ then $10 \times 2$ then $20 + 1$ then $21 ÷ 3$
then $7 \times 2$ then $14 - 2$ then $12 ÷ 3$ then $4 \times 2$ then $8 - 5$ and that is what we started
with at the beginning so we have finish the loop game.

Thank you for these well thought out solutions. Thank you also to those who pointed out that we had slipped up on day $1$ with an incorrect card.