Answer Key — with the working shown
Got something wrong? Don't worry — read the working and find where your thinking took a wrong turn.
How to use this page
Mark your work one question at a time. If you got it right — great! Move on. If you got it wrong — read the working in the green box, then have another go in your book before you peek at the next answer. Getting things wrong and figuring out why is how detectives learn.
🟢 Sheet 1 — Answers and working
A. Balance it
A1.5 + 3 = ☐
☐ = 8
5 + 3 = 8, so the box on the right must be 8.
A2.6 + 4 = ☐ + 2
☐ = 8
Step 1: work out the side you can. 6 + 4 = 10. → Step 2: now we need ☐ + 2 = 10. What plus 2 makes 10? 8.
Common slip-up: writing 10 in the box. Remember = means BALANCE, not "the answer is".
A3.9 + 1 = ☐ + 5
☐ = 5
Left side: 9 + 1 = 10. → Right side must also be 10. ☐ + 5 = 10, so ☐ = 5.
A4.7 + 7 = ☐ + 4
☐ = 10
Left: 7 + 7 = 14. → ☐ + 4 = 14, so ☐ = 10.
B. Continue the patterns
B1.2, 4, 6, 8, ☐, ☐
10, 12 · Step rule: + 2
Each number is 2 more than the last. 8 + 2 = 10, 10 + 2 = 12.
B2.3, 6, 9, 12, ☐, ☐
15, 18 · Step rule: + 3
These are the 3 times table! 12 + 3 = 15, 15 + 3 = 18.
B3.10, 15, 20, 25, ☐, ☐
30, 35 · Step rule: + 5
Counting in 5s. 25 + 5 = 30, 30 + 5 = 35.
B4.Harakeke Year 4 (Year 1 = 3, Year 2 = 5, Year 3 = 7)
Year 4 has 9 leaves
Each year the plant grows 2 more leaves. 7 + 2 = 9. (Drawing should show 9 leaves — any neat arrangement is fine.)
C. Crack the machines
C1.Machine rule + 3
OUTs: 5, 8, 13, 23
| IN | 2 | 5 | 10 | 20 |
|---|---|---|---|---|
| OUT | 5 | 8 | 13 | 23 |
2 + 3 = 5 ✓ (given), 5 + 3 = 8, 10 + 3 = 13, 20 + 3 = 23.
C2.Crack the rule from this table
The rule is + 5
Look at IN and OUT pairs: 1 → 6, 4 → 9, 7 → 12, 10 → 15. Each OUT is exactly 5 more than the IN. Test it on every clue — if it works for all four, you've cracked it.
Common slip-up: only checking one pair. The first pair (1 → 6) could also be × 6, but × 6 fails on 4 → 9. The right rule has to fit them ALL.
D. Missing number mysteries
D1.10 − ☐ = 6
☐ = 4
"10 take away something leaves 6." How far is it from 10 down to 6? Undo: 10 − 6 = 4. Check: 10 − 4 = 6 ✓
D2.12 + ☐ = 20
☐ = 8
Undo: 20 − 12 = 8. Check: 12 + 8 = 20 ✓
D3.☐ + 5 = 14
☐ = 9
Undo the + 5: 14 − 5 = 9. Check: 9 + 5 = 14 ✓
D4.☐ − 4 = 9
☐ = 13
This one is tricky! "Something take away 4 leaves 9." Undo the − 4 by + 4: 9 + 4 = 13. Check: 13 − 4 = 9 ✓
Common slip-up: writing 5 (because 9 − 4 = 5). Read it carefully — the mystery number is the BIG one we started with, before we took 4 away.
🟡 Sheet 2 — Answers and working
A. Trickier balance
A1.15 − 5 = ☐ + 4
☐ = 6
Left: 15 − 5 = 10. → ☐ + 4 = 10, so ☐ = 6.
A2.12 + 8 = ☐ + 13
☐ = 7
Left: 12 + 8 = 20. → ☐ + 13 = 20, so ☐ = 7.
A3.3 × 4 = ☐ + 2
☐ = 10
Left: 3 × 4 = 12. → ☐ + 2 = 12, so ☐ = 10.
A4.18 − ☐ = 6 + 5
☐ = 7
Step 1 — do the easy side first: 6 + 5 = 11. → Step 2: 18 − ☐ = 11. How far from 18 down to 11? 18 − 11 = 7. Check: 18 − 7 = 11 ✓
B. The shortcut rule
B1.Complete the kahikatea table
| Year | 1 | 2 | 3 | 4 | 5 | 10 |
|---|---|---|---|---|---|---|
| Branches | 4 | 7 | 10 | 13 | 16 | 31 |
4, 7, 10... up by 3 each time. 10 + 3 = 13, 13 + 3 = 16. Year 10 needs the shortcut (see B3).
B2.Step rule?
+ 3 each year
B3.Shortcut rule?
branches = year × 3 + 1
How to find it: the step is + 3, so the rule starts with "year × 3". For Year 1 that gives 3 — but we have 4 branches. So we need + 1. Rule: year × 3 + 1.
Test on Year 3: 3 × 3 + 1 = 10 ✓
Test on Year 3: 3 × 3 + 1 = 10 ✓
B4.Branches in Year 20?
61 branches
20 × 3 + 1 = 60 + 1 = 61. (No need to draw 20 trees!)
C. Two-step machines
C1.Crack this rule: 2 → 5, 3 → 7, 5 → 11, 10 → 21
Rule: × 2 then + 1
Detective thinking: 2 → 5 isn't + 3 or × 3 (×3 would give 6). What about two steps? 2 × 2 = 4, then 4 + 1 = 5 ✓. Test on every clue: 3 × 2 + 1 = 7 ✓, 5 × 2 + 1 = 11 ✓, 10 × 2 + 1 = 21 ✓. All four work!
C2.Same machine, IN = 8
OUT = 17
8 × 2 = 16, then 16 + 1 = 17.
C3.Same machine running backwards: OUT = 15
IN = 7
Undo the steps in REVERSE order (last in, first out). The machine did × 2 then + 1, so undo + 1 first, then undo × 2.
15 − 1 = 14 (undid the + 1). → 14 ÷ 2 = 7 (undid the × 2). Check forwards: 7 × 2 + 1 = 15 ✓
15 − 1 = 14 (undid the + 1). → 14 ÷ 2 = 7 (undid the × 2). Check forwards: 7 × 2 + 1 = 15 ✓
Common slip-up: doing the undoing in the wrong order — dividing by 2 first gives 7.5, which can't be right.
C4.Machine rule ÷ 2: IN 18 → ? OUT 7 → ?
18 → OUT 9 · OUT 7 ← IN 14
Forwards: 18 ÷ 2 = 9. Backwards (undo ÷ with ×): 7 × 2 = 14.
D. Missing numbers with × and ÷
D1.4 × ☐ = 20
☐ = 5
Undo × with ÷: 20 ÷ 4 = 5. Check: 4 × 5 = 20 ✓
D2.☐ × 5 = 45
☐ = 9
45 ÷ 5 = 9. Check: 9 × 5 = 45 ✓
D3.30 ÷ ☐ = 6
☐ = 5
"30 split into ☐ equal piles makes 6 in each." If 6 is the pile size, ☐ is how many piles: 30 ÷ 6 = 5. Check: 30 ÷ 5 = 6 ✓
D4.Tuna swims 42 km in 6 days, equal distance each day
7 km each day · sentence: 6 × ☐ = 42
Six equal days, total 42, so 6 × ☐ = 42. Undo × with ÷: 42 ÷ 6 = 7.
🔴 Sheet 3 — Answers and working
A. Solve for n
A1.n + 9 = 23
n = 14
Undo + 9 by − 9: 23 − 9 = 14. Check: 14 + 9 = 23 ✓
A2.n − 7 = 15
n = 22
Undo − 7 by + 7: 15 + 7 = 22. Check: 22 − 7 = 15 ✓
A3.4 × n = 36
n = 9
Undo × 4 by ÷ 4: 36 ÷ 4 = 9. Check: 4 × 9 = 36 ✓
A4.n ÷ 3 = 8
n = 24
Undo ÷ 3 by × 3: 8 × 3 = 24. Check: 24 ÷ 3 = 8 ✓
A5.2 × n + 1 = 17
n = 8
Two steps to undo, in reverse:
Step 1 — undo the + 1: 17 − 1 = 16. → So 2 × n = 16.
Step 2 — undo the × 2: 16 ÷ 2 = 8. → n = 8.
Check forwards: 2 × 8 + 1 = 16 + 1 = 17 ✓
Step 1 — undo the + 1: 17 − 1 = 16. → So 2 × n = 16.
Step 2 — undo the × 2: 16 ÷ 2 = 8. → n = 8.
Check forwards: 2 × 8 + 1 = 16 + 1 = 17 ✓
B. Write the rule with n
B1.Harakeke rule with n (where n = year)
leaves = n × 2 + 1
Same rule we had before, just with n instead of "year". Test on Year 3: 3 × 2 + 1 = 7 ✓
B2.Machine rule "× 3 then − 2" with n going in
OUT = n × 3 − 2
B3.Using OUT = n × 3 − 2: n = 10 and n = 100
n = 10 → 28 · n = 100 → 298
10 × 3 − 2 = 30 − 2 = 28. 100 × 3 − 2 = 300 − 2 = 298.
B4.Same machine: 28 came OUT. Write a puzzle with n and solve.
n × 3 − 2 = 28 · n = 10
Undo − 2 first: 28 + 2 = 30. → Now n × 3 = 30. Undo × 3: 30 ÷ 3 = 10. Check: 10 × 3 − 2 = 28 ✓
C. Reasoning like a mathematician
C1.Could the 3, 5, 7, 9... pattern ever have exactly 50 leaves?
No — never.
The rule is leaves = year × 2 + 1. year × 2 is always EVEN (anything × 2 is even). Even + 1 is always ODD. So every number in this pattern is odd: 3, 5, 7, 9, 11... 50 is even, so it can never appear. This is real mathematical reasoning — knowing without checking each one!
C2.☐ + ☐ + 4 = ☐ + 10 (both boxes hide the SAME number)
☐ = 6
Detective thinking: the left side has TWO boxes, the right side has ONE box. One box appears on both sides — we can "cancel" it (take one box off each side, the balance stays the same).
That leaves: ☐ + 4 = 10. → ☐ = 6. Check in the original: 6 + 6 + 4 = 16, and 6 + 10 = 16 ✓
That leaves: ☐ + 4 = 10. → ☐ = 6. Check in the original: 6 + 6 + 4 = 16, and 6 + 10 = 16 ✓
C3.2,500,000 ha of wetland; 10% left
250,000 ha left · 2,250,000 ha lost
10% means 1 in every 10 — so divide by 10: 2,500,000 ÷ 10 = 250,000 ha left.
Lost: 2,500,000 − 250,000 = 2,250,000 ha. That's the 90% destroyed.
Lost: 2,500,000 − 250,000 = 2,250,000 ha. That's the 90% destroyed.
For scale: 250,000 ha is about the size of Stewart Island. We lost an area nine times that.
D. Design your own
D1 & D2.Invented machines and patterns
Answers vary
How to check yours: a good two-step rule has to work on EVERY clue, not just one. If your partner found a different rule from yours but it still works on all your clues, then your puzzle had more than one possible answer — that's OK, it just means your puzzle needs an extra clue to be solvable. Real maths!