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

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.