Solve two puzzles step by step: rearrange a cross into two equal squares, and count routes through a letter diamond. Learn how to break a problem into manageable parts and check that your solution follows every rule.
Both puzzles become easier when we change how we look at them. You do not need to know the solutions yet—we will build each method one step at a time.
Your goal: Make exactly two straight cuts through the cross. Then rearrange all the pieces to form two squares of the same size.
Rules: Use every piece once. Leave no gaps or overlaps, and do not stretch any piece. Our guided solution uses sliding moves.
Our steps: Count the starting area in Mission 2, choose the cuts in Mission 3, and follow the reassembly in Mission 4. Counting unit squares helps us check that no area is lost.
Your goal: Find how many different routes lead from the top A to the bottom A.
Rules: Follow a drawn diagonal edge down exactly one row at each move. Do not go sideways, upward, or skip rows. Every complete route spells ABRACADABRA.
Two routes are different if they visit different positions in at least one row—even though they spell the same word.
Our steps: Trace one route in Mission 6, count all routes in Mission 7, and try other diamond sizes in Mission 8.
Start with Mission 1: practise choosing a useful strategy. After Puzzle A, Mission 5 gives you a routine for learning from attempts. Missions 9–10 let you check what you have learned from both puzzles.
A useful routine: Observe the conditions → Try an idea → Inspect the result → Revise if needed → Verify every condition. You can repeat these steps whenever you get stuck.
This is a strategy warm-up for the two puzzles above. Choose a useful next step for each situation; you are not being asked to solve either whole puzzle yet.
Decompose means break a complicated whole into simpler parts. For example, an L-shaped region may be easier to understand as two rectangles. Reassemble means put the parts together in a new arrangement.
Count locally means count ways to reach one position at a time. Suppose a position can be reached from only two neighbors: 3 routes reach the first neighbor and 2 reach the second. That gives 3 + 2 = 5 routes into this position. In Puzzle B, we will repeat this idea down the diamond.
When an attempt fails, inspect which rule it breaks and revise it. When it looks successful, verify every rule before calling it a solution.
Puzzle A, first step: Before cutting, find the area of the whole cross. Cutting and rearranging must preserve this total. For now, highlight and count the unit squares; the cutting comes next.
The cross is made from five congruent squares. Tap a square to highlight it; tap it again to unselect it. Each unit square has area 1 square unit. Highlighting helps you count—it does not cut or remove any part of the cross.
No squares selected yet. The whole cross still contains five unit squares.
Puzzle A, next step: The cross has area 5 square units. Now look for two straight cuts through O that will create pieces we can rearrange into two equal squares. Mission 4 will show how the chosen pieces fit together.
Choose two lines through O using the marked endpoints. The diagram updates as you choose each endpoint pair.
Puzzle A, finish and verify: The lines AC and BD from Mission 3 create the four colored pieces shown here. Follow this guided replay, then check that the pieces fill two equal squares without gaps or overlaps.
Pieces 1 and 2 can stay. Slide the opposite pieces into the dashed target outlines without rotating, flipping, stretching, or overlapping them. Each original unit square has area 1.
Pause between the puzzles: What would you do if your cut pieces did not fit? Practise a routine for inspecting and improving an attempt, then carry that habit into the letter puzzle.
Rehearse the displayed five-step routine once, using buttons 1 through 5 in that order. This is one useful model, not a requirement that every real investigation follow the same sequence. They are not a record of your own attempts; use the exploration notebook below to record what you tried and what changed.
Puzzle B, first step: Our goal is to count all different routes through this diamond. First trace just one complete route so you understand which moves are allowed.
Start at the top A. Follow a drawn diagonal edge down exactly one row to a neighboring letter. Do not move sideways, upward, or skip rows. Every row contains one fixed letter, so every complete legal route spells the same word.
Puzzle B, next step: Tracing one route tells us how to move. To count every route without losing track, replace each letter position with the number of routes that can reach it. The count at the bottom will answer the puzzle.
Try the same method on a new size: You have counted the routes through the original diamond. Keep the same downward-move rule and see what changes when the widest row has a different number of positions.
Correct all eight answers to complete the workshop. Areas are in original unit-square units; the angle is in degrees.
The worked example supplies the five-square cross construction, the ABRACADABRA letter diamond, the Pascal-style count ending at 252, and the advice to explore and revise rather than look immediately at an answer.
Added learning scaffolds: endpoint labels A–D, colored cut pieces, numerical area checks, interactive attempt history, accessibility controls, and the generalized path-width laboratory.