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Problem-Solving Strategies / LEVEL 1 · DIFFICULTY 1/5
Useful First Moves
Reverse processes, track invariants and force repetition.
Choose a node to explore its lesson.
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.
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.
A number is increased by 4, multiplied by 3, then decreased by 5 to give 31. Find the starting number.
- Reverse both the order of operations and each individual operation.
- .
- Forward verification: .
8
A number is increased by 5, multiplied by 3, then decreased by 5 to give 37. Find the starting number.
- Reverse both the order of operations and each individual operation.
- .
- Forward verification: .
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
A number is increased by 7, multiplied by 3, then decreased by 5 to give 49. Find the starting number.
A number is increased by 8, multiplied by 3, then decreased by 5 to give 55. Find the starting number.
A number is increased by 9, multiplied by 3, then decreased by 5 to give 61. Find the starting number.
A number is increased by 10, multiplied by 3, then decreased by 5 to give 67. Find the starting number.
A number is increased by 11, multiplied by 3, then decreased by 5 to give 73. Find the starting number.
A number is increased by 12, multiplied by 3, then decreased by 5 to give 79. Find the starting number.
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.
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.
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.
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.
A board begins with 9 lit lamps. Each move turns off exactly two currently lit lamps. After 2 moves, how many lamps remain lit?
- Each move subtracts an even number, so parity cannot change.
- Remaining=.
- 5 remain. An odd start can never reach zero under these moves.
5
A board begins with 11 lit lamps. Each move turns off exactly two currently lit lamps. After 3 moves, how many lamps remain lit?
- Each move subtracts an even number, so parity cannot change.
- Remaining=.
- 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
A board begins with 15 lit lamps. Each move turns off exactly two currently lit lamps. After 5 moves, how many lamps remain lit?
A board begins with 17 lit lamps. Each move turns off exactly two currently lit lamps. After 1 moves, how many lamps remain lit?
A board begins with 19 lit lamps. Each move turns off exactly two currently lit lamps. After 2 moves, how many lamps remain lit?
A board begins with 21 lit lamps. Each move turns off exactly two currently lit lamps. After 3 moves, how many lamps remain lit?
A board begins with 23 lit lamps. Each move turns off exactly two currently lit lamps. After 4 moves, how many lamps remain lit?
A board begins with 25 lit lamps. Each move turns off exactly two currently lit lamps. After 5 moves, how many lamps remain lit?
27 people enter a room. They can leave only in pairs. After 1 pairs leave, how many people remain?
29 people enter a room. They can leave only in pairs. After 2 pairs leave, how many people remain?
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.
Choose and carry out a method
Put at most objects in each of b boxes, giving . 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.
How many objects are needed to guarantee that some one of 4 boxes contains at least 3 objects, regardless of placement?
- Maximize the total while avoiding the target, then add one.
- At most objects fit with every box below the target.
- 9 guarantees the target; 8 can be evenly placed to avoid it.
9
How many objects are needed to guarantee that some one of 5 boxes contains at least 4 objects, regardless of placement?
- Maximize the total while avoiding the target, then add one.
- At most objects fit with every box below the target.
- 16 guarantees the target; 15 can be evenly placed to avoid it.
16
Common pitfalls
Possible mix-up: 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
How many objects are needed to guarantee that some one of 7 boxes contains at least 6 objects, regardless of placement?
How many objects are needed to guarantee that some one of 8 boxes contains at least 7 objects, regardless of placement?
How many objects are needed to guarantee that some one of 9 boxes contains at least 8 objects, regardless of placement?
How many objects are needed to guarantee that some one of 10 boxes contains at least 9 objects, regardless of placement?
How many objects are needed to guarantee that some one of 11 boxes contains at least 10 objects, regardless of placement?
How many objects are needed to guarantee that some one of 12 boxes contains at least 11 objects, regardless of placement?
A library has 13 genres. What is the smallest number of checked-out books that guarantees at least 12 belong to one genre?
A library has 14 genres. What is the smallest number of checked-out books that guarantees at least 13 belong to one genre?