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Before the next turn
Look one step ahead
Before choosing a move, ask what it will leave you able to do next.
A coastal path, a day to explore, and choices worth thinking through.
Mina and Li Wei have a day in a harbour town: a lookout to visit, sounds to record and a ferry to catch. Sprout comes along as their coach. At a fork, Mina spots a path with a very short first stretch.
Mina “Two minutes to the arch. That looks quick!”
CKSprout “What would choosing it leave you to do next?”
Li Wei “Let’s follow it in our minds before we turn.”
CKSTEM Thinking CompassStuck? Find your next move.
You don’t need the whole route yet. Start with one useful move, then notice what it tells you.
What do you know for certain? What are you trying to find?
After checking your steps against every clue, choose your own check:
Ready to explore.
The direction reflects your self-check. Use the lesson’s answer and reasoning to check further when you are ready.
The first stretch is only the beginning
The children have 8 minutes to reach the lookout. The map below shows all the allowed one-way paths and their exact travel times. They cannot turn back or take another path, and there are no extra waits.
Which first path, A or B, lets them reach the lookout within 8 minutes? Arriving at exactly 8 minutes would be in time. What is the total time for your chosen route?
Path A
Start → Arch2 min
Arch → Lookout9 min
Path B
Start → Bridge4 min
Bridge → Lookout3 min
Read stages in order. All allowed paths and exact times are shown; arrows allow only the shown direction.Try with the CKSTEM Thinking Compass
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Which first path, A or B, lets them reach the lookout within 8 minutes? Arriving at exactly 8 minutes would be in time. What is the total time for your chosen route?
The first stretch is only the beginning
The children have 8 minutes to reach the lookout. The map below shows all the allowed one-way paths and their exact travel times. They cannot turn back or take another path, and there are no extra waits.
Path A
Start → Arch2 min
Arch → Lookout9 min
Path B
Start → Bridge4 min
Bridge → Lookout3 min
Read stages in order. All allowed paths and exact times are shown; arrows allow only the shown direction.
Your answer
Enter your chosen journey’s time. Also name the route and explain why it fits.
The time fits. Check every stage of your chosen route against the eight-minute goal.
Break it into steps
Step 1 of 3
How long is the complete A journey?
\(2+9=11\) minutes. A correct sum can still miss the eight-minute goal.
Step 2 of 3
How long is the complete B journey?
\(4+3=7\) minutes.
Step 3 of 3
How long is the route you choose to meet the eight-minute goal?
B takes seven minutes, within eight. Also identify B and explain why the short first A stretch is not enough.
You have worked through these help steps. Return to Your answer when you are ready, and check every original clue.
A question to help me begin
Imagine that you have already taken one first path. Where are you now, and what still has to happen?
Show the answer and reasoning: the first turn
Answer: Choose path B. The complete journey takes 7 minutes; path A takes 11 minutes and misses the limit.
Li Wei “The arch leaves another nine minutes to the lookout. Two minutes isn’t the whole journey.”
CKSprout “How will you compare both choices against the actual goal?”
Mina “I’ll include what follows each first stretch. The bridge route starts slower but finishes sooner.”
Path A
2 + 9 = 11 minutes
Too late
Path B
4 + 3 = 7 minutes
Fits the eight-minute limit
Compare each complete journey with the same goal, not only its first stretch.
\[\begin{aligned}\text{A: }2+9&=11,\\\text{B: }4+3&=7.\end{aligned}\]
Check: both first stretches are allowed and fit within eight minutes. Only the complete B route reaches the lookout in time. The smallest next number was not the best choice.
Look one step ahead
Imagine taking a possible action. Then inspect the situation it creates: what could or must happen next? Compare that consequence with your goal before you commit.
This is useful when several moves are allowed but their later possibilities differ. A quick first gain can leave a poor next step. You can begin with a small trial in your mind or on a model; you need not know the whole solution to start learning from a choice.
“One step ahead” is a starting habit, not a promise that checking just one future move settles every problem.
Work backwards starts at a required finish to find earlier conditions. Here we start at a possible choice and ask what it leaves us able to do next. The two tools can help with different parts of one problem.
A missing stretch on the map
The route card for the lending hut has been smudged. Mina can read the first part of each route, but one travel time is missing.
The new map shows all allowed one-way paths to the hut, with three stretches per route. Known times are exact. The missing last stretch of A takes a positive whole number of minutes; its actual time is unknown. There are no other paths, extra waits or ways to turn back.
They must arrive within 7 minutes, including exactly 7. A's first two stretches take 1 minute each. B's three stretches take 3, 2 and 1 minutes.
Which route gives a fully checked journey in time, and how long does it take? Do these facts settle whether A would also work?
Path A
Start → A11 min
A1 → A21 min
A2 → Hut? min
Path B
Start → B13 min
B1 → B22 min
B2 → Hut1 min
The question mark is a missing actual time, a positive whole number of minutes. It is not zero. A possible time is not the actual time.Try with the CKSTEM Thinking Compass
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Which route gives a fully checked journey in time, and how long does it take? Do these facts settle whether A would also work?
A missing stretch on the map
The route card for the lending hut has been smudged. Mina can read the first part of each route, but one travel time is missing.
The new map shows all allowed one-way paths to the hut, with three stretches per route. Known times are exact. The missing last stretch of A takes a positive whole number of minutes; its actual time is unknown. There are no other paths, extra waits or ways to turn back.
They must arrive within 7 minutes, including exactly 7. A's first two stretches take 1 minute each. B's three stretches take 3, 2 and 1 minutes.
Path A
Start → A11 min
A1 → A21 min
A2 → Hut? min
Path B
Start → B13 min
B1 → B22 min
B2 → Hut1 min
The question mark is a missing actual time, a positive whole number of minutes. It is not zero. A possible time is not the actual time.
Your answer
Enter the checked journey’s time. Also decide what the facts establish about A.
The time fits. A’s actual last stretch is still unknown; the number does not settle A.
Break it into steps
Step 1 of 2
How many minutes do B’s first two stretches use?
\(3+2=5\) minutes so far; the journey is not yet complete.
Step 2 of 2
How long is the fully checked route to the hut?
\(3+2+1=6\) minutes for B. A’s actual last time stays unknown; this field does not decide whether A fits.
You have worked through these help steps. Return to Your answer when you are ready, and check every original clue.
Show the answer and reasoning: the incomplete map
Answer:B takes 6 minutes and is checked to fit. A is undecided: its missing stretch could make the journey fit or fail.
Mina “A uses only two minutes before the last stretch. That is less than B's first two stretches—but it isn't the whole journey.”
CKSprout “What can you establish, and what still depends on missing information?”
Li Wei “B is six minutes. For A, a five-minute last stretch would fit, but a nine-minute one would be late. Both are allowed by the card.”
B is checked
3 + 2 + 1 = 6 minutes
A is not settled
1 + 1 + ?
Last leg 5 → total 7, in time.
Last leg 9 → total 11, late.
The two A examples satisfy the missing-time rule and give different outcomes. Neither is asserted to be the actual time.
\[\begin{aligned}\text{B: }3+2+1&=6,\\\text{A could be: }1+1+5&=7,\\\text{or: }1+1+9&=11.\end{aligned}\]
Check: 5 and 9 are both positive whole-minute possibilities for the unreadable time. They give opposite conclusions about A, so its actual arrival cannot be determined. A short preview is useful information; it cannot fill a gap in the facts.
Two devices, one charging port
At the hut, they borrow equipment for a sound walk and photographs. The charging station has one timed port.
It is 12:00. The recorder needs an uninterrupted 20-minute charge for the sound walk leaving at 12:20, so it must be ready by then. The camera needs an uninterrupted 10-minute charge for photographs at 12:40, so it must be ready by then.
Charge one device at a time. There is no other port, pausing a charge or using a partly charged device. A charge finishing exactly at its deadline is ready in time.
Which charging order meets both needs? What are the earliest ready times in that order?
Recorder
20 minutes of continuous charging
Ready by 12:20
Camera
10 minutes of continuous charging
Ready by 12:40
One port is available at 12:00. The recorder is needed for the 12:20 walk; the camera for 12:40 photographs. No order is selected.Try with the CKSTEM Thinking Compass
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Which charging order meets both needs? What are the earliest ready times in that order?
Two devices, one charging port
At the hut, they borrow equipment for a sound walk and photographs. The charging station has one timed port.
It is 12:00. The recorder needs an uninterrupted 20-minute charge for the sound walk leaving at 12:20, so it must be ready by then. The camera needs an uninterrupted 10-minute charge for photographs at 12:40, so it must be ready by then.
Charge one device at a time. There is no other port, pausing a charge or using a partly charged device. A charge finishing exactly at its deadline is ready in time.
Recorder
20 minutes of continuous charging
Ready by 12:20
Camera
10 minutes of continuous charging
Ready by 12:40
One port is available at 12:00. The recorder is needed for the 12:20 walk; the camera for 12:40 photographs. No order is selected.
On the sound walk, Li Wei notes what the birds are doing as well as recording their calls. The same call can mean different things in different surroundings. Keeping that context gives them better evidence to interpret later; a recording alone is not an exact translation.
Your answer
Enter the earliest ready times using 24-hour h:mm. Check the order against both deadlines.
The earliest times fit. Check both deadlines and explain why this order works.
Break it into steps
Step 1 of 2
With the recorder first at 12:00, when is it ready?
12:00 plus 20 minutes is 12:20. It meets the sound-walk deadline.
Step 2 of 2
In that order, what is the earliest camera ready time?
12:20 plus 10 minutes is 12:30, before 12:40. Also justify why the other order fails.
You have worked through these help steps. Return to Your answer when you are ready, and check every original clue.
Show the answer and reasoning: the charging order
Answer: Charge the recorder first. The earliest ready times are 12:20 for the recorder and 12:30 for the camera. Both deadlines are met.
Mina “The camera charge finishes quickest. But if it goes first, the recorder can’t start until 12:10.”
CKSprout “What consequence does that have for the later task?”
Li Wei “The recorder would finish at 12:30, after the sound walk starts. Recorder first leaves enough time for both.”
Recorder first
Recorder12:00 → 12:20Ready in time
Camera12:20 → 12:30Ready in time
Camera first
Camera12:00 → 12:10Ready in time
Recorder12:10 → 12:30Late: needed by 12:20
These are the earliest ready times for each order, starting the next charge as soon as the port is free. Check both deadlines.
Check: recorder-first ends its 20 minutes at 12:20, then the camera’s ten minutes end at 12:30, before 12:40. Camera-first makes the recorder late. Those are the only two orders; adding idle time cannot make a late finish earlier.
The shortest next job is not always the right first job. The later need matters.
On the sound walk, Li Wei notes what the birds are doing as well as recording their calls. The same call can mean different things in different surroundings. Keeping that context gives them better evidence to interpret later; a recording alone is not an exact translation.
A game on the harbour bench
While the devices charge, Mina brings out ten reusable counters. Li Wei suggests a game with a choice at every turn.
Mina and Li Wei play with 10 counters in one pile. Mina moves first. Players alternate removing exactly 1, 2 or 3 counters, when that many are available. No passing, replacing counters or other moves. The player who takes the last counter wins.
Can Mina force a win? If so, how many counters should she take first, and what plan should she follow?
Ten counters, Mina to move. Each turn removes one, two or three; taking the last wins.Try with the CKSTEM Thinking Compass
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Can Mina force a win? If so, how many counters should she take first, and what plan should she follow?
A game on the harbour bench
While the devices charge, Mina brings out ten reusable counters. Li Wei suggests a game with a choice at every turn.
Mina and Li Wei play with 10 counters in one pile. Mina moves first. Players alternate removing exactly 1, 2 or 3 counters, when that many are available. No passing, replacing counters or other moves. The player who takes the last counter wins.
Ten counters, Mina to move. Each turn removes one, two or three; taking the last wins.
Your answer
Enter Mina’s first take. Her full plan must handle every legal reply.
The first take fits. Check every reply and both later rounds; a first number is not a complete strategy.
Break it into steps
Step 1 of 5
Imagine Li Wei starts with four and takes one. How many can Mina take to finish?
Four minus one leaves three; Mina can legally take them all.
Step 2 of 5
From four, Li Wei takes two. How many does Mina take to finish?
Two remain and Mina may take two.
Step 3 of 5
From four, Li Wei takes three. How many does Mina take to finish?
One remains; Mina takes it and wins.
Step 4 of 5
Starting from eight, a Li Wei/Mina pair removes four in total. How many remain?
\(8-4=4\). Check that Mina can choose a legal response for every Li Wei take, then repeat the rule.
Step 5 of 5
How many counters should Mina take first from ten?
Take two to leave eight. The number alone does not prove a win; verify all replies, both rounds and legal responses.
You have worked through these help steps. Return to Your answer when you are ready, and check every original clue.
Show the answer and reasoning: the counter game
Answer:Take 2 first, leaving 8. After each Li Wei turn, Mina takes enough to make their two takes total 4: she takes 3 after his 1, 2 after his 2, or 1 after his 3. This leaves him 4, then 0; Mina takes the last counter.
Mina “Taking three feels like a strong start. But that leaves seven, and Li Wei could take three and leave me four.”
CKSprout “What would you want your opponent's next turn to leave you able to do?”
Li Wei “If I had to play from four, I couldn't finish. Whichever amount I took, Mina could finish on her turn.”
Mina “Then I want to leave you four later. Can I leave eight now and return you to four after your next move?”
Look one step ahead
Choose a first move by the future it creates, not by how many counters it removes now. Leaving four to the other player lets you finish after their next turn. Leaving eight lets you arrange that same four-counter situation one round later. Mina takes two to reach eight, but that first number is not yet a winning plan: she must know how to respond when Li Wei chooses.
Casework
Separate Li Wei's possible takes into 1, 2 and 3. These are all legal replies from eight and from four. In each case, choose Mina's legal response so that the pair removes four counters. The response must work again after Li Wei makes a fresh choice; one cooperative sequence would not prove the promise.
Mina takes 2 → 8 remain
Li Wei faces 8
Every legal Li Wei take
Li Wei takes
Mina takes
Left after both
1
3
4
2
2
4
3
1
4
Li Wei faces 4
Every legal Li Wei take
Li Wei takes
Mina takes
Left after both
1
3
0
2
2
0
3
1
0
Li Wei may choose afresh in the second round. The same response covers all three options in each round, including all nine ordered pairs of replies.
\[1+3=2+2=3+1=4.\]
From eight, each complete round leaves four. From four, each complete round leaves zero, with Mina taking the last. Li Wei has three choices in each round: all nine ordered pairs of replies are covered by the same response rule.
Check the other starts: taking 1 leaves nine; Li Wei can take 1 and leave Mina eight. Taking 3 leaves seven; he can take 3 and leave Mina four. In either case, he can use the same response rule against her. Taking 2 is the only first move that forces a Mina win.
Looking ahead chooses useful future states. Casework makes the promised response complete and legal through both rounds. A first move without that response plan is not a forced win.
One more harbour recording
The afternoon sounds are changing. Mina and Li Wei would like to keep recording a little longer, but they need a departure plan they can justify.
It is 13:20. They may leave their recording spot now or later, and want to leave as late as possible with a fully checked plan to catch the ferry. It leaves at 14:00, but boarding closes at 13:55. Arriving at 13:55 is allowed.
Every route reaches bicycle parking first, followed by an exact 5-minute walk to boarding. Buses carry their cycles, have space and take exactly 12 minutes from their stop to parking. A bus can be boarded at its departure time, but not after it. There are no extra buses or delays. These are all four first choices; the times to each stop or junction are exact:
A: 4 minutes to a bus stop. Its only departures are 13:23 and 13:48.
B: 7 minutes to another bus stop. Its bus leaves at 13:38.
C: 6 minutes to a cycle junction. Every remaining route to parking is at least 5,000 metres.
D: 5 minutes to another junction. Every remaining route to parking is at least 4,000 metres.
On C and D, exact remaining distances and travel times are unknown. Cycling speed is at most 200 metres per minute; it may be lower. Every allowed C or D journey obeys these restrictions.
What is the latest time they can leave with a fully checked plan, and which first choice should they take? Explain what the facts show about the other choices.
Leave recording spot at a chosen time
Reach cycle parking
Walk 5 minutes
Board by 13:55
All four choices include the final walk. Keep recording until the latest departure that the given facts can justify.Try with the CKSTEM Thinking Compass
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What is the latest time they can leave with a fully checked plan, and which first choice should they take? Explain what the facts show about the other choices.
One more harbour recording
The afternoon sounds are changing. Mina and Li Wei would like to keep recording a little longer, but they need a departure plan they can justify.
It is 13:20. They may leave their recording spot now or later, and want to leave as late as possible with a fully checked plan to catch the ferry. It leaves at 14:00, but boarding closes at 13:55. Arriving at 13:55 is allowed.
Every route reaches bicycle parking first, followed by an exact 5-minute walk to boarding. Buses carry their cycles, have space and take exactly 12 minutes from their stop to parking. A bus can be boarded at its departure time, but not after it. There are no extra buses or delays. These are all four first choices; the times to each stop or junction are exact:
A: 4 minutes to a bus stop. Its only departures are 13:23 and 13:48.
B: 7 minutes to another bus stop. Its bus leaves at 13:38.
C: 6 minutes to a cycle junction. Every remaining route to parking is at least 5,000 metres.
D: 5 minutes to another junction. Every remaining route to parking is at least 4,000 metres.
On C and D, exact remaining distances and travel times are unknown. Cycling speed is at most 200 metres per minute; it may be lower. Every allowed C or D journey obeys these restrictions.
Leave recording spot at a chosen time
Reach cycle parking
Walk 5 minutes
Board by 13:55
All four choices include the final walk. Keep recording until the latest departure that the given facts can justify.
Your answer
Use 24-hour h:mm for the latest leaving time. Also justify the route and why no later start works.
The time fits. Check the whole route and explain why no other choice permits a later start.
Break it into steps
Step 1 of 7
What is the latest parking arrival that leaves time to board?
The walk must begin by 13:50.
Step 2 of 7
What is the latest useful bus departure to parking?
13:50 minus 12 minutes is 13:38. Compare this threshold with both actual timetables.
Step 3 of 7
Leaving now, how many minutes remain for C cycling after its first stretch and before parking?
From 13:26 to 13:50 there are 24 minutes.
Step 4 of 7
What is the maximum C cycling distance in that time?
\(24\times200=4{,}800\) metres, less than every allowed C route.
Step 5 of 7
What is the least possible D cycling time under its limits?
\(4{,}000\div200=20\) minutes is a lower bound, not D’s actual time.
Step 6 of 7
Even in its best case, what is D’s latest possible leaving time?
The full trip needs at least 30 minutes, so it must start by 13:25. This does not promise D works.
Step 7 of 7
What is the latest fully checked leaving time?
Leave at 13:31 for B. Check its complete journey and explain why none of the other choices can start later.
You have worked through these help steps. Return to Your answer when you are ready, and check every original clue.
Show the answer and reasoning: the final recording
Answer: Choose B and leave at 13:31. This reaches boarding at 13:55. A and C cannot work from 13:20 onwards. D might work from 13:20, but even its fastest permitted journey cannot allow departure after 13:25, so it cannot improve on B.
Mina “A has the shortest first stretch. Would that let us keep recording longest?”
CKSprout “What must a successful whole journey leave time for?”
Li Wei “The walk before boarding, and the bus before that. Let's find the useful departures before choosing when to leave.”
Work backwards
Start at the required finish to find a latest departure, rather than test a succession of guessed leaving times. Boarding by 13:55 means parking by 13:50. A twelve-minute bus ride must start by 13:38. This common threshold filters both timetables: A's 13:48 bus is too late, while B's 13:38 bus fits exactly. Undo each first stretch to find when the children would have to leave.
13:55 latest boarding
← undo 5-minute walk 13:50 latest parking
← undo 12-minute ride 13:38 latest useful bus
Reverse reasoning: teal and left arrows recover earlier thresholds from the required finish.
Choice A
Latest useful bus 13:23
Undo 4 minutes to its stop
Latest leave 13:19
Before now: unavailable
Choice B
Latest useful bus 13:38
Undo 7 minutes to its stop
Latest leave 13:31
Can leave after now
A timetable is not a continuous bus service. Use an actual departure at or before the common useful cutoff, then undo its first stretch.
A's useful 13:23 bus would require leaving at 13:19, before now. B's bus allows leaving at 13:31. But are those departures actually catchable, and could either cycling choice allow an even later start?
Look one step ahead
Imagine the state created by leaving at the proposed time. A short first stretch is useful only if it leaves a catchable connection. Follow the actual departure available on arrival, including the final walk; do not substitute the time of a bus that has already gone.
Choice A leaving 13:31
13:35 reach stop
13:48 catchable bus
14:00 parking
14:05 boarding
Choice B leaving 13:31
13:38 reach stop
13:38 catchable bus
13:50 parking
13:55 boarding
Forward check at the proposed latest departure. Amber stages show the actual connection, full ride and final walk; arrival exactly at departure is allowed.
Mina “At 13:31, B gets us to its stop exactly at 13:38. The bus takes us to parking at 13:50.”
CKSprout “What still needs to be established before calling this the latest start?”
Li Wei “That a cycling route can't let us start later. We don't have their exact lengths, but we do have limits.”
Bound the size
Use limits that apply to every allowed journey, without pretending they are actual travel times. Even leaving now, C reaches its junction at 13:26 and has at most 24 minutes before parking must be reached. At no more than 200 metres in each minute, it can cover at most 4,800 metres, less than every permitted C route. A later start leaves still less time.
Each filled block represents 200 metres. Even leaving now, C has only 24 cycling minutes. The fastest possible distance falls short of every allowed C route.
Equivalently, C needs at least \(5{,}000\div200=25\) minutes of cycling. Including its first stretch and the walk, even its best case needs \(6+25+5=36\) minutes. It would require leaving no later than 13:19.
D needs at least \(4{,}000\div200=20\) minutes of cycling. Its full trip therefore needs at least \(5+20+5=30\) minutes. Even its best case requires leaving by 13:25. Actual longer routes or lower speeds can only move that latest start earlier, never later than B's 13:31.
Mina “So D can't beat B. Does that also tell us whether D works if we leave now?”
CKSprout “What does the limit establish, and what is still missing?”
Li Wei “It doesn't give D's actual time. We can describe one allowed journey that fits now and another that doesn't.”
A possible D journey from 13:20
4,000 metres at 200 metres/minute
20 cycling minutes
Boarding 13:50
In time
A possible D journey from 13:20
6,000 metres at 200 metres/minute
30 cycling minutes
Boarding 14:00
Too late
Both obey D’s restrictions. They prove the given facts do not determine its actual arrival from 13:20; neither can beat B’s 13:31 departure.
Both examples obey D's restrictions. Its actual feasibility from 13:20 is undecided, but its inability to beat a 13:31 departure is proved. Those are different conclusions.
Check and prove latest: B leaving at 13:31 reaches the stop at 13:38, parking at 13:50 and boarding at 13:55, all under the stated rules. A requires departure by 13:19. Every C journey requires departure by 13:19 and every D journey by 13:25. B attains 13:31, and none of the four choices can leave later and still board.
How the ideas shared the work
Look one step ahead checked what connection and finish a proposed first choice actually leaves.
Work backwards found latest leaving times from the boarding cutoff and scheduled departures.
Bound the size ruled out every C trip and every D departure later than 13:25 without knowing actual route lengths.
A forward calculation is a valid check of a proposed departure. Reverse planning avoids guessing when to start; bounds settle whole families of unknown journeys. Explore Bound the size.
Your turn
Try a first choice in your mind. Follow what it leaves, then check the original goal before opening the answer.
Two buttons
A number machine starts at 4. Press Add 3 and Double once each, in either order, with no other operation.
To finish at 14, which button should come first?
CKSTEM Thinking CompassPlan and check your reasoning.
To finish at 14, which button should come first?
Two buttons
A number machine starts at 4. Press Add 3 and Double once each, in either order, with no other operation.
Start 4 · target 14
Add 3
Use once
Double
Use once
Choose an order for the two allowed buttons. No intermediate values are supplied.
You don’t need the whole route yet. Start with one useful move, then notice what it tells you.
What do you know for certain? What are you trying to find?
After checking your steps against every clue, choose your own check:
Ready to explore.
The direction reflects your self-check. Use the lesson’s answer and reasoning to check further when you are ready.
Start 4 · target 14
Add 3
Use once
Double
Use once
Choose an order for the two allowed buttons. No intermediate values are supplied.Show the answer and reasoning: the buttons
Answer: Press Add 3 first, then Double: \(4\to7\to14\).
Double first gives \(4\to8\to11\), which misses the target. Check: both allowed orders were considered and each button is used exactly once.
A changed game
There are 10 counters. You move first; players alternate taking exactly 1, 2 or 3 when available, with no passing or replacement. This time, taking the last counter loses.
How many counters should you take first to force a win, and what plan should you follow?
Ten counters. Players take one, two or three each turn; this time the last taker loses. Check what that changed goal means for your plan.Try with the CKSTEM Thinking Compass
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How many counters should you take first to force a win, and what plan should you follow?
A changed game
There are 10 counters. You move first; players alternate taking exactly 1, 2 or 3 when available, with no passing or replacement. This time, taking the last counter loses.
Ten counters. Players take one, two or three each turn; this time the last taker loses. Check what that changed goal means for your plan.
Your answer
Enter your first take. Also explain how your response plan handles the changed finish rule.
The first take fits. Check your response when the opponent changes their choice, and who takes the losing last counter.
Break it into steps
Step 1 of 3
For the last-taker-loses game, how many counters would you want to leave an opponent forced to play?
One forces the opponent to take the losing last counter.
Step 2 of 3
From nine, a complete opponent/you round removes four. How many remain?
\(9-4=5\). The next complete round can leave one for the opponent.
Step 3 of 3
How many counters should you take first from ten under the changed finish rule?
Take one to leave nine. Also justify all responses and why taking two or three fails against a permitted opponent plan.
You have worked through these help steps. Return to Your answer when you are ready, and check every original clue.
Show the answer and reasoning: last counter loses
Answer:Take 1 first, leaving 9. After each opponent take of 1, 2 or 3, take 3, 2 or 1 respectively. The pair removes four, returning 5 then 1 to your opponent. With one left, they must take it and lose.
Check: every response is legal from nine and from five. Taking 2 first leaves eight; the opponent can take 3 to leave you five and use the same response plan against you. Taking 3 first leaves seven; they can take 2 to leave you five. Changing only the finish rule changes the unique winning first move.
Pack what comes later
Two bags can carry 4 kg and 5 kg respectively. Three indivisible kits weigh 3 kg, 4 kg and 2 kg. All must fit without splitting, overloading or moving a packed kit. The 3 kg kit goes first.
Which bag should take it, and how will the other kits fit?
4 kg bag
Nothing packed yet
5 kg bag
Nothing packed yet
3 kg kit4 kg kit2 kg kit
Three indivisible kits. Place the 3 kg kit first and keep it there; do not overload either bag.Try with the CKSTEM Thinking Compass
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Which bag should take it, and how will the other kits fit?
Pack what comes later
Two bags can carry 4 kg and 5 kg respectively. Three indivisible kits weigh 3 kg, 4 kg and 2 kg. All must fit without splitting, overloading or moving a packed kit. The 3 kg kit goes first.
4 kg bag
Nothing packed yet
5 kg bag
Nothing packed yet
3 kg kit4 kg kit2 kg kit
Three indivisible kits. Place the 3 kg kit first and keep it there; do not overload either bag.
Your answer
Enter the chosen bag’s capacity. Also explain how all three whole kits fit.
The capacity fits. Check the full packing, with no splitting, moving or overloading.
Break it into steps
Step 1 of 4
If the three-kilogram kit goes into the four-kilogram bag, how much capacity remains there?
\(4-3=1\) kg remains. Neither remaining whole kit fits that space.
Step 2 of 4
How much do the other two kits weigh together?
\(4+2=6\) kg, too much for the other five-kilogram bag.
Step 3 of 4
If the three-kilogram kit goes into the five-kilogram bag, how much capacity remains?
\(5-3=2\) kg remains, enough for the two-kilogram kit.
Step 4 of 4
What is the capacity of the bag that should receive the three-kilogram kit?
Use the five-kilogram bag for 3+2; the four-kilogram kit fits the other bag. Also explain the packing; capacity alone does not describe it.
You have worked through these help steps. Return to Your answer when you are ready, and check every original clue.
Show the answer and reasoning: the bags
Answer: Put the 3 kg kit in the 5 kg bag, then add the 2 kg kit there. The 4 kg kit goes in the 4 kg bag.
If the 3 kg kit went into the 4 kg bag, only 1 kg would remain there. Neither remaining kit fits that space, and together they weigh 6 kg, too much for the other bag. Check: the successful loads are \(3+2=5\) kg and 4 kg, with every whole kit used once.
Look back
Ask what a choice leaves possible, what still has to happen, and which consequences are known. Check every relevant reply when someone else chooses. If your preview stops before a condition that can change the result, keep looking.
Mina and Li Wei arrive with a plan they can explain. They have time to enjoy the crossing—and to listen to the harbour again.
Continue with Bound the size: a useful limit can settle a question before you know every detail.
Mina “Two minutes to the arch. That looks quick!”
CKSprout “What would choosing it leave you to do next?”
Li Wei “Let’s follow it in our minds before we turn.”