Work Backwards

Undo the add $9$ first, $41-9=32$. Then undo the doubling, $32÷2=\boxed{{\displaystyle 16}}$.

Check it forward, $16$ doubled is $32$, and $32+9=41$.

Undo the add $7$, $52-7=45$. Undo the multiply by $5$, $45÷5=9$. Undo the subtract $4$, $9+4=\boxed{{\displaystyle 13}}$.

Forward, $13-4=9$, times $5$ is $45$, plus $7$ is $52$.

Uma sold for $32$ at a profit of $8$, so she paid $32-8=24$. That $24$ is what Theo got for the card, and it left him down $5$, so he paid $24+5=\boxed{{\displaystyle 29}}$.

A seller with a profit paid less than the sale price, so undoing a profit subtracts and undoing a loss adds.

Rewind day two. Before the $6$ liters went in the barrel held $21-6=15$, and before losing half it held $30$. So day one ended at $30$.

Rewind day one the same way, $30-6=24$, then double to get $\boxed{{\displaystyle 48}}$.

Inside each day the pour happens last, so it is the first thing you undo.

Working back from day $10$, day $9$ held half a jar, and day $8$ held half of that, which is one quarter.

$$\boxed{{\displaystyle 8}}$$

Day $10$ is the only amount you are given, so stepping backward from it beats hunting for the starting size.

Each number is the sum of the two before it, so $b+19=31$, giving $b=12$. One step earlier, $a+12=19$, so $a=\boxed{{\displaystyle 7}}$.

Forward, the list reads $7,12,19,31,50$, and each term is the sum of the two before it.

Rewinding one hour divides by $3$, and four hours pass between $1$ and $5$.

$$405÷3÷3÷3÷3=405÷81=\boxed{{\displaystyle 5}}$$

Forward check, $5,15,45,135,405$, one tripling per hour.

Rewind the second pour. It doubled A to $36$, so A held $18$ before it, and the $18$ that arrived came from B, which held $36+18=54$.

Rewind the first pour. It doubled B to $54$, so B held $27$, and that $27$ came from A, which started with $18+27=\boxed{{\displaystyle 45}}$.

Forward, $45$ and $27$ become $18$ and $54$, then $36$ and $36$.

Rewind from $11$. Add back the exit fee to get $14$, double to get $28$, add back the entry fee to get $35$, and double again to get $\boxed{{\displaystyle 70}}$.

Forward, half of $70$ leaves $35$, the entry fee leaves $28$, half leaves $14$, and the exit fee leaves $11$.

Adding back the $9$ gives $48$, and dividing by $4$ gives $\boxed{{\displaystyle 12}}$. Forward, $12\times 4-9=39$.

From $45$, dividing by $5$ gives $9$, multiplying by $3$ gives $27$, and subtracting $8$ gives $\boxed{{\displaystyle 19}}$. Forward, $(19+8)÷3\times 5=45$.

Sam paid $87+9=96$. Rosa received that $96$ at a $12$-token profit, so she had paid $96-12=\boxed{{\displaystyle 84}}$.

Rewind the second sibling, $14-2=12$, and doubling gives $24$. Rewind the first, $24-2=22$, and doubling gives $\boxed{{\displaystyle 44}}$.

Forward, $44\to 22+2=24\to 12+2=14$.

Backward, $96,000\to 48,000\to 24,000\to 12,000$, three halvings, so $\boxed{{\displaystyle 3}}$ days earlier.

Rewinding, before $13$ sits $21-13=8$, and before that sits $13-8=5$. The list runs $5,8,13,21,34$, and four places before the $34$ is $\boxed{{\displaystyle 5}}$.

Two rewinds give $486÷3=162$ and $162÷3=\boxed{{\displaystyle 54}}$.

Undo the moves in reverse, $6-2=4$, then $4+8=12$, then $12-5=\boxed{{\displaystyle 7}}$.

Forward, $7+5-8+2=6$.

Adding the $4$ back gives $30$, and doubling gives $\boxed{{\displaystyle 60}}$. Forward, half of $60$ is $30$, minus $4$ is $26$.

Rewinding one link means adding $1$ and halving. From $41$, the third number is $(41+1)÷2=21$, the second is $(21+1)÷2=11$, and the first is $(11+1)÷2=\boxed{{\displaystyle 6}}$.

Forward, $6\to 11\to 21\to 41$.