Reasoning

Thinking in steps

1. Build the instructions

   Turn a goal into a sequence of moves, then follow what each command actually does.

   Read the guide and try it

   Keep position and direction separate

   A marker follows the commands you give it. Forward changes its position. A turn changes where it faces. Planning both makes the result easier to predict.

   1. Name the goal and check the starting direction. Left and right are relative to the marker, not the screen.
   2. Build a short sequence. A repeated forward command still visits every square, so it cannot jump over a barrier.
   3. Step through the plan. Compare the final square and facing with the goal, then change any command that needs attention.

   Try the idea

   A marker faces south. It follows Turn left, Forward, Turn right, Forward. Where does it finish relative to its starting square?

   1. One square west and one south, facing south.
   2. One square east and one south, facing south.
   3. One square east and one north, facing north.

   Show answer

   One square east and one south, facing south. Left from south points east. Move east once, turn right to face south, then move south once. Neither turn changes the square.

   Keep in mind: Read each command from the current state. A correct destination can still have the wrong final direction.

   Open practice

2. Find it in fewer checks

   Use an ordered collection to rule out whole groups of places after each check.

   Read the guide and try it

   Narrow the places that can still match

   This method depends on the values being ordered from smallest to largest. The gaps may vary and values may repeat. Without a known order, one opened drawer cannot rule out a whole side.

   1. Check a middle position in the remaining range. Choose by position, not by guessing the numerical halfway value.
   2. Compare its value with the target. Keep only the higher side, keep only the lower side, or stop at an equal value.
   3. Repeat within the smaller range. Any equal value is a valid find here; an empty range proves the target is absent.

   Try the idea

   Seven drawers are ordered from smallest to largest, with uneven gaps and possible repeats. You want 47. Drawer 4 contains 52. Which drawers can you rule out immediately?

   1. Drawers 4 through 7.
   2. Drawer 4 only.
   3. Drawers 1 through 4.

   Show answer

   Drawers 4 through 7. Drawer 4 is 52, and every drawer to its right is 52 or larger. None can contain 47. Drawers 1 through 3 remain possible.

   Keep in mind: Exclude places because the order proves they cannot match. Choosing a middle position keeps the two remaining possibilities nearly balanced.

   Open practice

3. Sort with a strategy

   Build an ordered section while keeping track of every card, including repeated values.

   Read the guide and try it

   Make one comparison at a time

   Two useful methods organize the work differently. Insertion finds a place for the next card. Selection finds the smallest card still waiting. Both preserve the whole collection.

   1. For insertion, compare the next card with its left neighbor in the ordered section. Move left until it fits.
   2. For selection, keep a smallest-so-far candidate. Compare it with every remaining card before placing it.
   3. Keep every card. Equal values may appear in either order unless a separate rule requires their original order.

   Try the idea

   During a selection pass, you have compared cards with values 8, 5 and 5, keeping one 5 as the smallest candidate. One card remains unchecked. What should happen next?

   1. Place the candidate 5 and start the next pass.
   2. Start the pass again because two cards were equal.
   3. Compare the remaining card with the candidate 5.

   Show answer

   Compare the remaining card with the candidate 5. Finish the pass before placing its smallest card. Keep 5 if the remaining value is larger, replace it if smaller, and accept either candidate if equal.

   Keep in mind: A candidate is only the smallest seen so far. An ordered result must include every original card exactly once.

   Open practice

4. Repair the command chain

   Inspect a plan that gives the wrong output and test a small change to make it work.

   Read the guide and try it

   Judge the result of the repaired plan

   A command acts on the result of the previous command. A useful repair may replace, move, add or remove a step. More than one repair can produce the required output.

   1. Read the starting row, target and edit limit. Run the original chain one step at a time to see what it changes.
   2. Try a small repair and run it again from the same starting row. Compare the new output with the target.
   3. Check the whole result and the edit limit. Different command orders may work, but do not assume all commands can be exchanged.

   Try the idea

   Start with 4, 1, 6 and aim for 8, 3, 6. Plan A reverses the row, then adds 2 to every number. Plan B adds 2 to every number, then reverses the row. Which plan reaches the target?

   1. Plan A only.
   2. Both plans.
   3. Plan B only.

   Show answer

   Both plans. Both produce 8, 3, 6. Adding the same amount to every number can happen before or after a rearrangement. That does not mean every pair of commands can change order.

   Keep in mind: A repair is successful when its executed output meets the goal within the stated limits. It need not match one suggested command chain.

   Open practice

5. Solve a smaller problem

   Break a tower move into smaller tasks, then stop breaking it down when one disk can move directly.

   Read the guide and try it

   Give the spare peg a job

   To move a tower, first clear its largest disk. Use the spare peg for the smaller tower, move the largest disk, then bring the smaller tower onto it. Every actual move still takes just one disk.

   1. Name the starting peg, destination and spare peg for the current task. Their roles can change in a smaller task.
   2. Move the smaller tower to the spare peg, move the largest disk to the destination, then move the smaller tower onto it.
   3. When a task has one disk, move it directly if legal. That smaller task is complete; continue the rest of the larger plan.

   Try the idea

   While following a larger tower plan, you reach the task “Move one disk from B to A.” That disk is on top of B, and A holds only larger disks. How should this smaller task end?

   1. Find a still smaller tower to move away first.
   2. Move the disk from B to A and finish the whole puzzle.
   3. Move the disk from B to A, then continue the larger plan.

   Show answer

   Move the disk from B to A, then continue the larger plan. One disk is the stopping case. Move it directly, complete that task, and return to the remaining work in the larger plan.

   Keep in mind: Use the same three-part plan on smaller towers. Stop dividing a task when one legal disk move can complete it.

   Open practice

6. Follow the rule

   Check each condition, then choose the branch it leads to.

   Read the guide and try it

   Read the full condition before choosing.

   Compare boundary values carefully. Then combine the checks and follow only the first matching branch.

   1. Test a single condition, including its boundary.
   2. AND needs both checks. OR needs at least one, including both.
   3. Follow branches in order and stop at the first match.

   Try the idea

   A card has number 5 and is marked. The rule asks for 5 or more AND marked. Does it match?

   1. Yes. Both checks pass.
   2. No. The number must be above 5.
   3. No. AND means only one check may pass.

   Show answer

   Yes. Both checks pass. “5 or more” includes 5. AND requires both checks, and this card passes both.

   Keep in mind: Compare boundary values carefully. Then combine the checks and follow only the first matching branch.

   Open practice

7. Manage the waiting line

   Follow each arrival and removal. The rule decides who leaves next.

   Read the guide and try it

   Know which end can change.

   Update the collection after every operation. A new arrival never jumps to the front of a queue, but goes to the top of a stack.

   1. A queue adds at the back and takes from the front. This is first in, first out (FIFO).
   2. A stack adds and takes at the top. This is last in, first out (LIFO).
   3. Before taking an item, check whether the collection is empty. If it is, report that nothing can be removed.

   Try the idea

   A and then B join a queue. A leaves, then C joins. Who leaves next?

   1. C, the latest arrival.
   2. B, the earliest item still waiting.
   3. A, because it arrived first.

   Show answer

   B, the earliest item still waiting. A has already left. B is at the front; C joined at the back.

   Keep in mind: Update the collection after every operation. A new arrival never jumps to the front of a queue, but goes to the top of a stack.

   Open practice