🔎 Congruency and Similarity, SGrade 8Reading: grades 7-9ExponentsSquare rootsPythagorean theoremOrder of operationsTwo-step equations

The Great Blueprint Heist

A rogue geometer has altered the scale factors of Isometric City's core blueprints! If the measurements aren't restored to their perfectly congruent proportions, the entire city's towers will collapse like a house of cards. The Chief Architect needs your help to find the culprit before the countdown ends.

Suspects

Possible suspects

Cross off a row as each clue rules it out. On screen, tap a row.

21 of 21 still in play

Name	Scaling Tool	Secondary Gear	Hand Dominance	Hair Style	Vulnerability
Pascal	Dilation Beam	Proportional Compass	Left-handed stylus use	Slicked silver braids	Uncalibrated scales
Fibonacci	Reflection Mirror	Ratio Caliper	Left-handed stylus use	Spiky copper locks	Uncalibrated scales
Mandelbrot	Shear Stretcher	Vector Thruster	Left-handed stylus use	Slicked silver braids	Uncalibrated scales
Riemann	Rotation Pivot	Similarity Scanner	Right-handed stylus use	Neon blue buzzcut	Asymmetrical mirrors
Bolyai	Rotation Pivot	Similarity Scanner	Right-handed stylus use	Neon blue buzzcut	Static distortion fields
Euclid	Translation Slide	Vector Thruster	Left-handed stylus use	Neon blue buzzcut	Static distortion fields
Escher	Dilation Beam	Ratio Caliper	Right-handed stylus use	Neon blue buzzcut	Uncalibrated scales
Hilbert	Translation Slide	Grid Overlay HUD	Left-handed stylus use	Slicked silver braids	Uncalibrated scales
Noether	Reflection Mirror	Proportional Compass	Left-handed stylus use	Spiky copper locks	Static distortion fields
Hypatia	Dilation Beam	Proportional Compass	Right-handed stylus use	Neon blue buzzcut	Static distortion fields
Archimedes	Rotation Pivot	Vector Thruster	Left-handed stylus use	Spiky copper locks	Static distortion fields
Leibniz	Dilation Beam	Proportional Compass	Left-handed stylus use	Neon blue buzzcut	Static distortion fields
Kepler	Translation Slide	Proportional Compass	Right-handed stylus use	Slicked silver braids	Uncalibrated scales
Cantor	Reflection Mirror	Vector Thruster	Left-handed stylus use	Spiky copper locks	Static distortion fields
Euler	Translation Slide	Ratio Caliper	Left-handed stylus use	Slicked silver braids	Uncalibrated scales
Lovelace	Reflection Mirror	Similarity Scanner	Left-handed stylus use	Spiky copper locks	Asymmetrical mirrors
Lobachevsky	Translation Slide	Similarity Scanner	Left-handed stylus use	Slicked silver braids	Asymmetrical mirrors
Gauss	Reflection Mirror	Ratio Caliper	Right-handed stylus use	Slicked silver braids	Uncalibrated scales
Fermat	Rotation Pivot	Proportional Compass	Left-handed stylus use	Slicked silver braids	Static distortion fields
Thales	Rotation Pivot	Ratio Caliper	Left-handed stylus use	Slicked silver braids	Uncalibrated scales
Descartes	Shear Stretcher	Proportional Compass	Left-handed stylus use	Slicked silver braids	Asymmetrical mirrors

Clue 1

Exponents

We cracked the encryption on the thief's tablet. The passcode length grew exponentially based on the blueprint's scale dimensions. Solve the exponent chain to unlock the system files.

Solve each problem to learn its letter. Then write that letter in every box that shows the same number, and read the secret clue.

400

225

216

225

441

400

128

484

196

400

225

196

216

225

400

169

400

128

196

216

225

400

225

5^2 =T

2^4 =R

6^3 =U

21^2 =P

15^2 =S

2^3 =H

2^7 =C

22^2 =D

20^2 =E

13^2 =V

7^2 =N

14^2 =O

0/12 letters found

Clue 2

Square roots

To restore a shrunken gallery, we have to find the actual side length of its perfectly square floor plan. Taking the square root of the scaled area will unlock the security locker.

Solve each problem to learn its letter. Then write that letter in every box that shows the same number, and read the secret clue.

sqrt(100) =T

sqrt(25) =L

sqrt(361) =W

sqrt(121) =D

sqrt(81) =F

sqrt(144) =K

sqrt(196) =A

sqrt(16) =Y

sqrt(64) =I

sqrt(4) =E

sqrt(289) =S

sqrt(256) =R

sqrt(324) =H

sqrt(169) =C

sqrt(49) =N

0/15 letters found

Clue 3

Pythagorean theorem

The culprit ran diagonally across the digital coordinate grid. We must calculate the hypotenuse of their scaled path to find where they dropped a crucial clue.

Solve each problem to learn its letter. Then write that letter in every box that shows the same number, and read the secret clue.

legs 24 and 32, hypotenuse =T

legs 16 and 30, hypotenuse =A

legs 24 and 70, hypotenuse =R

legs 9 and 12, hypotenuse =U

legs 25 and 60, hypotenuse =P

legs 24 and 45, hypotenuse =C

legs 6 and 8, hypotenuse =F

legs 18 and 24, hypotenuse =L

legs 10 and 24, hypotenuse =Y

legs 3 and 4, hypotenuse =B

legs 15 and 36, hypotenuse =S

legs 20 and 48, hypotenuse =D

legs 21 and 72, hypotenuse =E

legs 8 and 15, hypotenuse =I

legs 40 and 75, hypotenuse =N

legs 9 and 40, hypotenuse =H

0/16 letters found

Clue 4

Order of operations

The Rescaler left a booby-trapped formula combining several scale ratios, additions, and divisions. Run the order of operations perfectly to defuse the trap.

Solve each problem to learn its letter. Then write that letter in every box that shows the same number, and read the secret clue.

108

101

100

108

100

101

100

108

101

11 × 9 - 20 =A

98 + 5 × 2 =S

1 + 2 × 5 =O

20 + 9 × 9 =L

29 + 7 × 3 =D

76 + 6 × 4 =I

41 + 9 × 6 =V

11 × 4 - 2 =E

8 + 6 × 6 =B

9 × 12 - 10 =C

33 + 7 × 3 =F

7 + 4 × 5 =N

8 × 10 - 25 =K

8 × 9 - 6 =T

9 × 5 - 5 =W

31 + 9 × 7 =R

0/16 letters found

Clue 5

Two-step equations — the last clue

A laser beam's target height depends on the original scale factor times the base height plus a constant offset. Solve this two-step equation to trace the culprit's laser beam.

First solve each problem. Then find each answer in the list below and cross that sentence out. One sentence will be left — that's exactly what the villain did!

Step 1 — solve these

3x + 9 = 21, x = ?

3x + 12 = 33, x = ?

5x + 5 = 20, x = ?

5x + 4 = 44, x = ?

5x + 5 = 10, x = ?

2x + 12 = 24, x = ?

4x + 4 = 48, x = ?

5x + 9 = 69, x = ?

4x + 9 = 17, x = ?

6x + 11 = 65, x = ?

4x + 1 = 41, x = ?

Step 2 — cross out the sentence with each answer

1.The villain slid the city hall grid, then recalibrated the proportional compasses.

2.The villain tilted the structural pillars, then scrambled the grid overlay system.

3.The villain rotated the suspension bridge, then locked the ratio calipers.

4.The villain slid the city hall grid, then scrambled the grid overlay system.

5.The villain slid the city hall grid, then locked the ratio calipers.

6.The villain tilted the structural pillars, then locked the ratio calipers.

7.The villain shrank the central monument, then recalibrated the proportional compasses.

8.The villain shrank the central monument, then scrambled the grid overlay system.

9.The villain rotated the suspension bridge, then scrambled the grid overlay system.

10.The villain slid the city hall grid, then fired up the vector thrusters.

11.The villain shrank the central monument, then locked the ratio calipers.

12.The villain flipped the skyline reflections, then fired up the vector thrusters.

Solve it

Name the culprit

Choose the one suspect left standing.

Answer Key — The Great Blueprint Heist

Culprit: Euler (Translation Slide · Ratio Caliper · Left-handed stylus use · Slicked silver braids · Uncalibrated scales)

Clue 1 (Exponents): "THE SUSPECT DOES NOT USE VECTOR THRUSTERS"
5^2 = 25 (T) · 2^4 = 16 (R) · 6^3 = 216 (U) · 21^2 = 441 (P) · 15^2 = 225 (S) · 2^3 = 8 (H) · 2^7 = 128 (C) · 22^2 = 484 (D) · 20^2 = 400 (E) · 13^2 = 169 (V) · 7^2 = 49 (N) · 14^2 = 196 (O)
Clue 2 (Square roots): "THE YARDSTICK WAS HELD IN THE LEFT HAND"
sqrt(100) = 10 (T) · sqrt(25) = 5 (L) · sqrt(361) = 19 (W) · sqrt(121) = 11 (D) · sqrt(81) = 9 (F) · sqrt(144) = 12 (K) · sqrt(196) = 14 (A) · sqrt(16) = 4 (Y) · sqrt(64) = 8 (I) · sqrt(4) = 2 (E) · sqrt(289) = 17 (S) · sqrt(256) = 16 (R) · sqrt(324) = 18 (H) · sqrt(169) = 13 (C) · sqrt(49) = 7 (N)
Clue 3 (Pythagorean theorem): "THE THIEF IS TRIPPED BY UNCALIBRATED SCALES"
legs 24 and 32, hypotenuse = 40 (T) · legs 16 and 30, hypotenuse = 34 (A) · legs 24 and 70, hypotenuse = 74 (R) · legs 9 and 12, hypotenuse = 15 (U) · legs 25 and 60, hypotenuse = 65 (P) · legs 24 and 45, hypotenuse = 51 (C) · legs 6 and 8, hypotenuse = 10 (F) · legs 18 and 24, hypotenuse = 30 (L) · legs 10 and 24, hypotenuse = 26 (Y) · legs 3 and 4, hypotenuse = 5 (B) · legs 15 and 36, hypotenuse = 39 (S) · legs 20 and 48, hypotenuse = 52 (D) · legs 21 and 72, hypotenuse = 75 (E) · legs 8 and 15, hypotenuse = 17 (I) · legs 40 and 75, hypotenuse = 85 (N) · legs 9 and 40, hypotenuse = 41 (H)
Clue 4 (Order of operations): "A STRAND OF SLICKED SILVER BRAID WAS LEFT"
11 × 9 - 20 = 79 (A) · 98 + 5 × 2 = 108 (S) · 1 + 2 × 5 = 11 (O) · 20 + 9 × 9 = 101 (L) · 29 + 7 × 3 = 50 (D) · 76 + 6 × 4 = 100 (I) · 41 + 9 × 6 = 95 (V) · 11 × 4 - 2 = 42 (E) · 8 + 6 × 6 = 44 (B) · 9 × 12 - 10 = 98 (C) · 33 + 7 × 3 = 54 (F) · 7 + 4 × 5 = 27 (N) · 8 × 10 - 25 = 55 (K) · 8 × 9 - 6 = 66 (T) · 9 × 5 - 5 = 40 (W) · 31 + 9 × 7 = 94 (R)
Clue 5 (Two-step equations): surviving statement (box 5) → Euler
3x + 9 = 21, x = 4 · 3x + 12 = 33, x = 7 · 5x + 5 = 20, x = 3 · 5x + 4 = 44, x = 8 · 5x + 5 = 10, x = 1 · 2x + 12 = 24, x = 6 · 4x + 4 = 48, x = 11 · 5x + 9 = 69, x = 12 · 4x + 9 = 17, x = 2 · 6x + 11 = 65, x = 9 · 4x + 1 = 41, x = 10

About this mystery

Grade level: Grade 8 math
Math topics: Exponents, Square roots, Pythagorean theorem, Order of operations, Two-step equations
Reading level: grades 7-9
Theme: The Great Blueprint Heist — set in Isometric City
Format: Interactive online & printable worksheet with answer key
Solution: One unique, verified culprit among 21 suspects

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