Substitution Arithmetic
A custom symbol is just a recipe waiting to be substituted.
1 Explore — try these first
Try before you watch. Pick a level below and give the problem an honest try on paper first — wrong turns and all. Then open the video to see the trick. Every level rides one habit: Code, substitute, compute — every custom symbol bows to the same three moves.
L0 · Repeated Addition
A new button ★ is defined by two examples: 3 ★ 2 = 3 + 3 and 4 ★ 3 = 4 + 4 + 4. Using the same pattern, what is 5 ★ 4?
L1 · Arithmetic Trail
A two-symbol code a ★ b ★ c is shown by two examples: 1 ★ 2 ★ 3 = 1 + 3 + 5 and 4 ★ 3 ★ 4 = 4 + 7 + 10 + 13. Using the same pattern, what is 5 ★ 2 ★ 6?
L2 · Inside Out
A diamond rule says a ◇ b = 3a − 2b. Using this rule, what is 5 ◇ (4 ◇ 1)?
L3 · Substitute and Solve
A blend rule says a ◈ b = (a + b) ÷ (1 + ab). If 2 ◈ x = 4 ⁄ 7, find x.
L4 · Decode the Rule
A box rule satisfies 1 ☆ 2 = 7, 2 ☆ 1 = 8, and 1 ☆ 3 = 10. Assuming a ☆ b = ma + nb + c, find the value of x for which x ☆ 5 = 20.
L5 · Decode and Chain
A rule satisfies 1 ☆ 1 = 4, 1 ☆ 2 = 7, and 2 ☆ 1 = 6. Assuming a ☆ b = ab + ma + nb, compute 3 ☆ (2 ☆ 1).
2 Learn — watch the solutions
Gave each one a real try? Now watch the trick. (Stuck is fine — that's the point.)
L0 · The Star Button — Repeated Addition
Peek the trick — Repeated Addition
When a custom symbol expands as one number added a few times, it is just multiplication in disguise. Read p ★ q as p added q times, and rewrite it as p × q.
L1 · The Two-Symbol Code — Arithmetic Trail
Peek the trick — Arithmetic Trail
A three-number code start ★ step ★ count secretly spells an arithmetic sequence. Build the trail with the step, then sum it with the Gauss formula count × (first + last) ÷ 2.
L2 · The Diamond Rule — Inside Out
Peek the trick — Inside Out
When a custom symbol is nested inside brackets, never apply it to the outside first. Solve the inner bracket, collapse it to a single number, then substitute outward and apply the rule again.
L3 · The Blend Equation — Substitute and Solve
Peek the trick — Substitute and Solve
When a coded equation has an unknown and the rule is a fraction, substitute the rule, cross-multiply, and let the messy expression collapse into a simple linear equation.
L4 · Hidden Rule, Three Clues — Decode the Rule
Peek the trick — Decode the Rule
When the rule is hidden but three example equations are given, assume the linear template ma + nb + c. Three examples make three equations in m, n, c — solve the system, then the rule is yours.
L5 · Bilinear Disguise — Decode and Chain
Peek the trick — Decode and Chain
Same decode move as Level 4, but the rule now hides a product term. Assume a ☆ b = ab + ma + nb, recover m and n from the examples, then chain the rule through nested brackets.
3 Master — practice on your own
Print the practice sheet and solve without the videos. Check your answers at the back — if one is wrong, the answer key names the trick so you know exactly which video to rewatch.
Download fresh practice problems PDF
