Open cases · crack them in order

The case files

Four cases, one of each type. Decode before you solve. Stuck? Open the green drawer first and knock three times. Only peek at the gold drawer once you have given it a real go.

Each case has three drawers. Green = the Three Knocks nudge. Purple = a smaller case to warm up on. Gold = the full solution. Try to crack it before the gold drawer.

CASE 01 🪣 Tank Case Spice2 chillis
Kelcy is saving for a concert ticket that costs $180. She already has $48 saved, and she puts away $12 every week. How many weeks until she has enough?

Before you solve: read it twice, underline the question, circle the three numbers and their units.

👀 Stuck? Knock three times

Knock, knock, knock: What am I doing? Why? Is it helping?

This is a Tank Case, so it has a start value and a steady rate. Which number is the head start she already has? Which number happens again every week? And what is the target she is aiming for? Label them before you do any sums.

🔍 Show me a smaller case

Warm up on tiny numbers first: You want a $20 toy. You already have $6, and you save $2 a week. How many weeks?

Notice how you used the $6 head start before dividing. Now do your real case exactly the same way.

✅ Show the full solution

Decode. Start value (already saved) = $48. Rate = $12 each week. Target = $180. Question = how many weeks, call it w.

Translate. Amount after w weeks is 48 + 12w, and we want it to reach 180:

48 + 12w = 180

  1. Take off the head start she already has: 180 − 48 = 132 left to save.
  2. Each week adds $12, so divide: 132 ÷ 12 = 11.
  3. So w = 11 weeks.
🌶️ The trap It is tempting to just do 180 ÷ 12 = 15 weeks. That forgets the $48 she already has, so it gives too many weeks. The head start is the clue most people miss.
👃 Sniff test After 11 weeks: 48 + 12 × 11 = 48 + 132 = $180. That lands exactly on the target, and 11 weeks is a believable amount of saving. It makes sense.
Case closed ✓
📓 Case Notes
  • What kind of case was this, and how did you know?
  • What got you unstuck, the head start or the rate?
  • This is secretly y = mx + c. What was the m, and what was the c?
CASE 02 ↩️ Backwards Case Spice2 chillis
A goblin is playing a number game. She says: "I thought of a number, I doubled it, then I took away 7, and I ended up with 23." What number was she thinking of?

Before you solve: which operation did she do first, and which did she do second? You will undo them in the opposite order.

👀 Stuck? Knock three times

Knock, knock, knock: What am I doing? Why? Is it helping?

A Backwards Case is like rewinding a video. She went forwards: double, then take away 7. To find the start, run it backwards: undo the last thing first. The opposite of "take away 7" is...? The opposite of "double" is...?

🔍 Show me a smaller case

Warm up: A goblin doubled a number, took away 1, and got 9. What was it?

Undo backwards: add the 1 back (9 + 1 = 10), then halve (10 ÷ 2 = 5). Check: 2 × 5 − 1 = 9. Now do your real case the same way.

✅ Show the full solution

Decode. Call the mystery number x. She doubled it (2x), took away 7 (2x − 7), and got 23.

2x − 7 = 23

Run it backwards (undo in the opposite order):

  1. Undo "take away 7" by adding 7 back: 23 + 7 = 30, so 2x = 30.
  2. Undo "double" by halving: 30 ÷ 2 = 15.
  3. So x = 15.
🌶️ The trap If the puzzle were written "7 less than double a number is 23", that is still 2x − 7 = 23, NOT 7 − 2x. "Less than" flips the order. Also watch the undo order: add the 7 back before you halve, not after.
👃 Sniff test Put 15 back into the story: double it to get 30, take away 7 to get 23. That is exactly what she ended on. It makes sense.
Case closed ✓
📓 Case Notes
  • Why did you add the 7 back before halving, and not the other way round?
  • Where is the "less than" trap, and how do you avoid it?
  • What does a Backwards Case feel like, so you spot the next one?
CASE 03 📦 Container Case Spice2 chillis
A goblin keeps her pencils in a cube-shaped box. The box holds exactly 64 cm³. She wants to cover the whole outside in sticker paper. How much paper does she need?

Before you solve: it gives you the volume, but it asks for something else. What is it actually asking for, and what is the hidden first step?

👀 Stuck? Knock three times

Knock, knock, knock: What am I doing? Why? Is it helping?

"Cover the whole outside" means the total surface area, not the volume. But to find the area of the faces, you first need the length of one side. The volume can give you that. What number, multiplied by itself three times, makes 64?

🔍 Show me a smaller case

Warm up: A cube holds 8 cm³. How long is one side? (What number times itself three times is 8? It is 2, because 2 × 2 × 2 = 8.)

Then: one square face is 2 × 2 = 4 cm², and a cube has 6 faces, so 6 × 4 = 24 cm². Now do your real case the same way.

✅ Show the full solution

Decode. Shape = cube. Given: volume = 64 cm³. Asked: total surface area (the paper to cover all 6 faces).

  1. Hidden step, find the side. For a cube, volume = side × side × side. We need the number that cubes to 64: 4 × 4 × 4 = 64, so the side = 4 cm.
  2. One face. Each face is a square: 4 × 4 = 16 cm².
  3. All six faces. A cube has 6 faces: 6 × 16 = 96 cm².

She needs 96 cm² of sticker paper.

🌶️ The trap Two traps here. First, do not stop at the volume or hand back 64. Second, the answer is an area, so the unit is cm², not cm³. Wrong unit, wrong answer.
👃 Sniff test Side 4 cm cubes back to 64 cm³, which matches. The answer is in cm² because it is paper covering a surface. 96 cm² is a sensible size for a little box. It makes sense.
Case closed ✓
📓 Case Notes
  • What was the hidden first step, and what tipped you off that you needed it?
  • How do you stop yourself mixing up cm² and cm³?
  • What makes a Container Case a Container Case?
CASE 04 🔀 Crossover Case Spice3 chillis
Two goblins start saving on the same day. Mossy already has $50 and saves $5 each week. Pip already has $20 and saves $8 each week. After how many weeks will they have exactly the same amount, and how much will each have then?

A spicy one. This is a two-part question, so make sure you answer both parts. "The same amount" is the big clue.

👀 Stuck? Knock three times

Knock, knock, knock: What am I doing? Why? Is it helping?

Each goblin is a Tank Case on her own: a start value plus a steady rate. Write an expression for Mossy's money after w weeks, and one for Pip's. "The same amount" tells you what to do with those two expressions. (Hint: it is the "=" clue from the translation table.)

🔍 Show me a smaller case

Warm up with friendlier numbers: Mossy has $10 and saves $1 a week. Pip has $4 and saves $3 a week. When do they have the same amount?

Set them equal: 10 + w = 4 + 3w. That gives 6 = 2w, so w = 3 weeks (each has $13). Now scale up to the real numbers the same way.

✅ Show the full solution

Decode. Mossy after w weeks: 50 + 5w. Pip after w weeks: 20 + 8w. "The same amount" means set them equal.

50 + 5w = 20 + 8w

  1. Get the w terms together. Take 5w off both sides: 50 = 20 + 3w.
  2. Take 20 off both sides: 30 = 3w.
  3. Divide by 3: w = 10 weeks.
  4. Part two, how much? Use either goblin: Mossy = 50 + 5 × 10 = $100.

After 10 weeks they each have $100.

🌶️ The trap Two traps. Do not stop at w = 10 and forget the second part (how much). And when you collect the w terms, take the smaller one (5w) off both sides so you keep w positive. This is also a Crossover on a graph: two lines meeting at one point.
👃 Sniff test Check Pip too: 20 + 8 × 10 = $100. Both land on $100, so the crossover is real. Pip saves faster, so it makes sense she catches up to Mossy's head start. It makes sense.
Case closed ✓
📓 Case Notes
  • How did "the same amount" tell you to set the two expressions equal?
  • Why take 5w off both sides and not 8w?
  • If you drew this as two lines, what would the crossover point be?
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