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Problem-Solving Strategies / LEVEL 1 · DIFFICULTY 1/5

Useful First Moves

Reverse processes, track invariants and force repetition.

3 stages · 24 practice problems · two 6-question assessment forms

Choose an island to read its lesson.

  1. MINI QUEST PS 1.1Work BackwardRead the lesson
  2. MINI QUEST PS 1.2Track an InvariantRead the lesson
  3. MINI QUEST PS 1.3Force a RepetitionRead the lesson
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STAGE PS 1.1 · 8 PRACTICE PROBLEMS · 2 NEW-CONTEXT APPLICATIONS

Work Backward

Goal: Understand and apply work backward.

Before you begin: Algebra, geometry, counting and the logarithm definition. Later stages combine subjects.

Understand the idea

When a process has a known final state and reversible steps, reversing it can expose the starting state directly. The last forward operation must be the first one undone.

inverse of composition reverses operation order

Choose and carry out a method

List the forward operations in order, then apply their inverses in reverse order. Keep parentheses around grouped intermediate quantities.

Check the reasoning

Run the original process forward from your recovered value. This catches errors that a plausible-looking inverse expression may hide.

WORKED EXAMPLE 1

A number is increased by 4, multiplied by 3, then decreased by 5 to give 31. Find the starting number.

  1. Reverse both the order of operations and each individual operation.
  2. (31+5)/3-4=8.
  3. Forward verification: (8+4)·3-5=31.

8

WORKED EXAMPLE 2

A number is increased by 5, multiplied by 3, then decreased by 5 to give 37. Find the starting number.

  1. Reverse both the order of operations and each individual operation.
  2. (37+5)/3-5=9.
  3. Forward verification: (9+5)·3-5=37.

9

Common pitfalls

Possible mix-up: Undo the first operation first.

Start with the last operation because it acts nearest the known final state.

Possible mix-up: A correct numerical answer alone explains the method.

State the governing relationship and check the conditions described above.

Explain it to yourself

Explain why reversing the operation order matters.

Preview the eight practice prompts
  1. A number is increased by 7, multiplied by 3, then decreased by 5 to give 49. Find the starting number.
  2. A number is increased by 8, multiplied by 3, then decreased by 5 to give 55. Find the starting number.
  3. A number is increased by 9, multiplied by 3, then decreased by 5 to give 61. Find the starting number.
  4. A number is increased by 10, multiplied by 3, then decreased by 5 to give 67. Find the starting number.
  5. A number is increased by 11, multiplied by 3, then decreased by 5 to give 73. Find the starting number.
  6. A number is increased by 12, multiplied by 3, then decreased by 5 to give 79. Find the starting number.
  7. A game adds 13 bonus points to a score, triples the result, then charges 5 points. The final score is 85. Find the original score. New context
  8. A game adds 14 bonus points to a score, triples the result, then charges 5 points. The final score is 91. Find the original score. New context
Open stage PS 1.1 in the student workspace →

STAGE PS 1.2 · 8 PRACTICE PROBLEMS · 2 NEW-CONTEXT APPLICATIONS

Track an Invariant

Useful preparation: Work Backward

Goal: Understand and apply track an invariant.

Before you begin: Work Backward

Understand the idea

An invariant is a property preserved by every allowed move. Subtracting two changes a count but preserves its parity, which can rule out some target states immediately.

(n−2k) mod 2=n mod 2

Choose and carry out a method

Identify the quantity changed by a move and compare it modulo two before and after. Compute the requested remaining count, then inspect the parity conclusion.

Check the reasoning

A preserved property gives a necessary condition for reachability, not always a sufficient one. State exactly what it rules out.

WORKED EXAMPLE 1

A board begins with 9 lit lamps. Each move turns off exactly two currently lit lamps. After 2 moves, how many lamps remain lit?

  1. Each move subtracts an even number, so parity cannot change.
  2. Remaining=9-2·2.
  3. 5 remain. An odd start can never reach zero under these moves.

5

WORKED EXAMPLE 2

A board begins with 11 lit lamps. Each move turns off exactly two currently lit lamps. After 3 moves, how many lamps remain lit?

  1. Each move subtracts an even number, so parity cannot change.
  2. Remaining=11-2·3.
  3. 5 remain. An odd start can never reach zero under these moves.

5

Common pitfalls

Possible mix-up: An invariant means the entire number never changes.

The number may change while a property such as parity stays fixed.

Possible mix-up: A correct numerical answer alone explains the method.

State the governing relationship and check the conditions described above.

Explain it to yourself

Why can an odd positive count never reach zero by removing pairs only?

Preview the eight practice prompts
  1. A board begins with 15 lit lamps. Each move turns off exactly two currently lit lamps. After 5 moves, how many lamps remain lit?
  2. A board begins with 17 lit lamps. Each move turns off exactly two currently lit lamps. After 1 moves, how many lamps remain lit?
  3. A board begins with 19 lit lamps. Each move turns off exactly two currently lit lamps. After 2 moves, how many lamps remain lit?
  4. A board begins with 21 lit lamps. Each move turns off exactly two currently lit lamps. After 3 moves, how many lamps remain lit?
  5. A board begins with 23 lit lamps. Each move turns off exactly two currently lit lamps. After 4 moves, how many lamps remain lit?
  6. A board begins with 25 lit lamps. Each move turns off exactly two currently lit lamps. After 5 moves, how many lamps remain lit?
  7. 27 people enter a room. They can leave only in pairs. After 1 pairs leave, how many people remain? New context
  8. 29 people enter a room. They can leave only in pairs. After 2 pairs leave, how many people remain? New context
Open stage PS 1.2 in the student workspace →

STAGE PS 1.3 · 8 PRACTICE PROBLEMS · 2 NEW-CONTEXT APPLICATIONS

Force a Repetition

Useful preparation: Track an Invariant

Goal: Understand and apply force a repetition.

Before you begin: Track an Invariant

Understand the idea

The pigeonhole principle guarantees a crowded category when there are too many objects to keep every category below a target. A sharp threshold needs both a guarantee and an avoiding example.

minimum guarantee=b(r−1)+1

Choose and carry out a method

Put at most r−1 objects in each of b boxes, giving b(r−1). One additional object forces some box to reach r.

Check the reasoning

Show that the smaller total really can avoid the target by filling every box equally. This proves minimality, not just sufficiency.

WORKED EXAMPLE 1

How many objects are needed to guarantee that some one of 4 boxes contains at least 3 objects, regardless of placement?

  1. Maximize the total while avoiding the target, then add one.
  2. At most 4·(3-1)=8 objects fit with every box below the target.
  3. 9 guarantees the target; 8 can be evenly placed to avoid it.

9

WORKED EXAMPLE 2

How many objects are needed to guarantee that some one of 5 boxes contains at least 4 objects, regardless of placement?

  1. Maximize the total while avoiding the target, then add one.
  2. At most 5·(4-1)=15 objects fit with every box below the target.
  3. 16 guarantees the target; 15 can be evenly placed to avoid it.

16

Common pitfalls

Possible mix-up: b·r objects are necessary.

The target is forced as soon as the avoiding capacity is exceeded by one.

Possible mix-up: A correct numerical answer alone explains the method.

State the governing relationship and check the conditions described above.

Explain it to yourself

Construct an arrangement showing that one fewer object does not suffice.

Preview the eight practice prompts
  1. How many objects are needed to guarantee that some one of 7 boxes contains at least 6 objects, regardless of placement?
  2. How many objects are needed to guarantee that some one of 8 boxes contains at least 7 objects, regardless of placement?
  3. How many objects are needed to guarantee that some one of 9 boxes contains at least 8 objects, regardless of placement?
  4. How many objects are needed to guarantee that some one of 10 boxes contains at least 9 objects, regardless of placement?
  5. How many objects are needed to guarantee that some one of 11 boxes contains at least 10 objects, regardless of placement?
  6. How many objects are needed to guarantee that some one of 12 boxes contains at least 11 objects, regardless of placement?
  7. A library has 13 genres. What is the smallest number of checked-out books that guarantees at least 12 belong to one genre? New context
  8. A library has 14 genres. What is the smallest number of checked-out books that guarantees at least 13 belong to one genre? New context
Open stage PS 1.3 in the student workspace →
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