Diagrams where paths agree

A square on the integers has “add $3$” across the top and “multiply by $2$” down the right; down the left it has “multiply by $2$” and across the bottom “add $6$”. Does it commute?

Weekdays are numbered $0$ (Monday) to $6$ (Sunday). Today is weekday $4$ (Friday). Using the square “add $100$ days, then take the weekday = take the weekday, then add $2$ and take the remainder on division by $7$”, which weekday number will it be in $100$ days?

Which of these functions from $\Z$ to $\Z$ preserve addition, $f(a + b) = f(a) + f(b)$? Choose all that apply.

Kwame tests a square whose two routes are $x \mapsto (x + 1)^2$ and $x \mapsto x^2 + 1$. At $x = 0$ both give $1$, so he says the square commutes. Is he right?

A rug measures $2$ m by $3$ m. What is its area in square centimetres?

A square has “add a delivery fee of $300$ shillings” across the top and “divide by $125$” (shillings to dollars) down both sides. Which arrow across the bottom makes the square commute?

Two squares are placed side by side, sharing a middle arrow. You are told that the outer rectangle commutes. What can you conclude?

A function $f : \Z \to \Z$ preserves addition, and $f(3) = 12$. What is $f(5)$?

Which Lean statement says that converting metres to centimetres, $x \mapsto 100x$, preserves addition?

Casting out nines checks a multiplication with remainders on division by $9$. Write digit_sum(n) (the sum of the decimal digits of a whole number n) and nines_check(a, b, claimed), which returns True when digit_sum(a) * digit_sum(b) and digit_sum(claimed) leave the same remainder on division by $9$, and False otherwise. Can the check ever pass for a wrong answer?