The Lumenary · Grades 6–8

Diagrams where paths agree

Name: ______________________

Date: ______________________

You will learn to

Check whether the two routes around a square of functions agree for every input, and refute it with a single input where they differ
Express statements such as “doubling then adding 2 equals adding 1 then doubling” as diagrams whose paths agree
Recognise maps that preserve addition, $f(a + b) = f(a) + f(b)$ , such as converting metres to centimetres, as commuting squares, and show that $x \mapsto x + 1$ does not
Prove that two commuting squares placed side by side form a commuting rectangle, and show that the converse fails

1.
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?
- A.Yes, for every input
- B.Only for $x = 0$
- C.No, the routes never agree
- D.Only for positive inputs
2.
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?
3.
Which of these functions from $\Z$ to $\Z$ preserve addition, $f(a + b) = f(a) + f(b)$ ? Choose all that apply.
- A. $x \mapsto 7x$
- B. $x \mapsto x - 2$
- C. $x \mapsto 0$ (everything goes to $0$ )
- D. $x \mapsto x^2$
- E. $x \mapsto -x$
4.
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.No: at $x = 1$ the routes give $4$ and $2$
- B.Yes: one input is enough
- C.Yes: $0$ is a special input, so it counts for all
- D.It is impossible to tell without checking every integer
5.
A rug measures $2$ m by $3$ m. What is its area in square centimetres?
6.
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?
- A.add $2.40$ dollars
- B.add $300$ dollars
- C.multiply by $2.40$
- D.add $125 \div 300 \approx 0.42$ dollars
7.
Two squares are placed side by side, sharing a middle arrow. You are told that the outer rectangle commutes. What can you conclude?
- A.Nothing about the two squares: either of them may fail to commute
- B.Both squares commute
- C.The left square commutes, but the right one may not
- D.At least one of the two squares commutes
8.
A function $f : \Z \to \Z$ preserves addition, and $f(3) = 12$ . What is $f(5)$ ?
9.
Which Lean statement says that converting metres to centimetres, $x \mapsto 100x$ , preserves addition?
- A.∀ a b : Int, 100 * (a + b) = 100 * a + 100 * b
- B.∀ a b : Int, 100 * (a * b) = (100 * a) * (100 * b)
- C.∃ a b : Int, 100 * (a + b) = 100 * a + 100 * b
- D.∀ a : Int, 100 * (a + a) = 100 * a + 100 * a
10.
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?
Computational — Practice

Answer key — Diagrams where paths agree

1.
Across then down: $x \mapsto x + 3 \mapsto 2(x + 3) = 2x + 6$ . Down then across: $x \mapsto 2x \mapsto 2x + 6$ . These are equal for every integer by the distributive law, so the square commutes.
2.
Because $100 = 14 \cdot 7 + 2$ , the square says the weekday moves on by $2$ : $(4 + 2) \bmod 7 = 6$ , a Sunday. (Check the long way: if today is day $4$ , then day $104$ has remainder $104 - 14 \cdot 7 = 6$ .)
3.
$x \mapsto 7x$ , $x \mapsto 0$ and $x \mapsto -x$ preserve addition: they are multiplication by $7$ , $0$ and $-1$ . $x \mapsto x - 2$ fails at $a = b = 0$ , and $x \mapsto x^2$ fails at $a = b = 1$ . This matches Theorem 4: the addition-preserving functions are exactly the multiplications.
4.
At $x = 1$ : $(1 + 1)^2 = 4$ but $1^2 + 1 = 2$ . So the square does not commute. In fact $(x + 1)^2 - (x^2 + 1) = 2x$ , so the routes agree only at $x = 0$ — exactly the one input Kwame tried.
5.
The sides are $200$ cm and $300$ cm, so the area is $200 \cdot 300 = 60\,000$ cm². Converting the area $6$ m² by “ $\times 100$ ” would give the wrong answer $600$ : the square “multiply, then convert” = “convert both, then multiply” does not commute, because $100(ab) \neq (100a)(100b)$ . Since $1$ m² $= 100 \cdot 100 = 10\,000$ cm², the right factor for areas is $10\,000$ .
6.
Across then down: $p \mapsto (p + 300) \div 125 = \tfrac{p}{125} + 2.4$ . Down then across with bottom arrow “add $d$ ”: $p \mapsto \tfrac{p}{125} + d$ . These agree for every $p$ exactly when $d = 2.40$ dollars — the converted fee.
7.
The converse of pasting is false. If the arrows at the right-hand end of both outer routes send everything to $0$ , then both outer routes give $0$ and the rectangle commutes, no matter what the squares do. So a commuting rectangle tells us nothing about the individual squares. (The lesson's experiment found $4160$ fillings on $\{0, 1\}$ where the rectangle commutes but the left square does not.)
8.
$12 = f(3) = f(1) + f(1) + f(1) = 3f(1)$ , so $f(1) = 4$ . By Theorem 4, $f(x) = f(1) \cdot x = 4x$ for every integer, so $f(5) = 20$ .
9.
∀ a b : Int, 100 * (a + b) = 100 * a + 100 * b — this is what PreservesAdd (fun x => 100 * x) unfolds to, and cm_preserves_add proves it from the distributive law.
10.
```
def digit_sum(n):
    total = 0
    while n > 0:
        total += n % 10
        n //= 10
    return total


def nines_check(a, b, claimed):
    return (digit_sum(a) * digit_sum(b)) % 9 == digit_sum(claimed) % 9
```
The check works because the square “multiply, then take the remainder on division by $9$ ” = “take remainders, then multiply and take the remainder” commutes, and a number and its digit sum leave the same remainder. It catches $1071$ but passes $1801$ : any wrong answer with the right remainder slips through, for example one with two digits swapped. A failed check proves an error; a passed check is only evidence.