The problem states the number system is less than four, meaning it could be base 2 or base 3. The rewritten pin "12 20 22 10" contains the digit '2'. Since base 2 (binary) only uses digits 0 and 1, th

The problem states the number system is less than four, meaning it could be base 2 or base 3. The rewritten pin "12 20 22 10" contains the digit '2'. Since base 2 (binary) only uses digits 0 and 1, the number system cannot be base 2. Therefore, the number system must be base 3 (ternary), which uses digits 0, 1, and 2.
The number system used is base 3.

Base 3 is used because the rewritten pin contains the digit '2', which is not allowed in base 2, and the problem states the base is less than four.

Each pair of digits in the rewritten pin represents an original base-10 digit. We convert each base-3 number to base 10:

• First digit: $12_3 = (1 \times 3^1) + (2 \times 3^0) = 3 + 2 = 5_{10}$
• Second digit: $20_3 = (2 \times 3^1) + (0 \times 3^0) = 6 + 0 = 6_{10}$
• Third digit: $22_3 = (2 \times 3^1) + (2 \times 3^0) = 6 + 2 = 8_{10}$
• Fourth digit: $10_3 = (1 \times 3^1) + (0 \times 3^0) = 3 + 0 = 3_{10}$ The original ATM pin is $\boxed{\text{5683}}$.

The nurse checks every 2 hours, which is $2 \times 60 = 120$ minutes.
The doctor checks every 4.5 hours, which is $4.5 \times 60 = 270$ minutes.

We need to find LCM(120, 270).
Prime factorization of 120: $120 = 2^3 \times 3 \times 5$
Prime factorization of 270: $270 = 2 \times 3^3 \times 5$
LCM(120, 270) = $2^3 \times 3^3 \times 5 = 8 \times 27 \times 5 = 1080$ minutes.

1080 minutes = $\frac{1080}{60} = 18$ hours.
They last checked together at 9:30 am.
Adding 18 hours to 9:30 am:
9:30 am + 12 hours = 9:30 pm
9:30 pm + 6 hours = 3:30 am (the next day)
They will check on him again at the same time at $\boxed{\text{3:30 am}}$.

Total hours in a day = 24 hours.
Resting time = $\frac{3}{8} \times 24 \text{ hours} = 9 \text{ hours}$.
Eating time = $\frac{1}{6} \times 24 \text{ hours} = 4 \text{ hours}$.

Remaining time after resting and eating = $24 - 9 - 4 = 11 \text{ hours}$.
Time spent having a healthy diet = $\frac{2}{3} \times 11 \text{ hours} = \frac{22}{3} \text{ hours}$.

Time spent visiting the hospital = Remaining time - Healthy diet time
Time spent visiting the hospital = $11 - \frac{22}{3} = \frac{33}{3} - \frac{22}{3} = \frac{11}{3}$ hours per day.

Total hours per week = $\frac{11}{3} \text{ hours/day} \times 7 \text{ days/week} = \frac{77}{3}$ hours/week.
$\frac{77}{3} \text{ hours} = 25 \frac{2}{3} \text{ hours}$.
He spends $\boxed{\frac{77}{3} \text{ hours or } 25 \frac{2}{3} \text{ hours}}$ visiting the hospital in a week.