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Signals from the sea
Use logical reasoning
What must be true—and what have we not found out yet?
A listening exhibit is coming together at the coast. The team checks what its clues really establish.
Rowan lifts one earcup. There is a whole soundscape to explore at the ocean discovery centre. Mina is helping prepare a programme so visitors can hear different recordings. CKRobo is their coach.
Rowan “That sound makes me think of a boat. But thinking of something doesn’t prove it.”
CKRobo “What information would let you support your explanation?”
Mina “We can check the recording’s clues. And say what we still don’t know.”
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.
Combine two clues
The four available recordings last 30, 40, 60 and 75 seconds.
The exhibit needs one that is longer than 35 seconds and shorter than 70 seconds. It must also use the entire recording in complete 15-second sections, with no cutting, skipping, repeating or added silence.
Which recording length works? Explain why each of the other three lengths fails at least one rule.
30 s40 s60 s75 s
Length window
Longer than 35 s
Shorter than 70 s
Whole sections
15 s in each section
Use the entire clip
The cards give the available lengths. Every stated condition applies to the chosen recording.Try with the CKSTEM Thinking Compass
Interactive checking needs JavaScript. The complete question and worked answer remain available.
Which recording length works? Explain why each of the other three lengths fails at least one rule.
Combine two clues
The four available recordings last 30, 40, 60 and 75 seconds.
The exhibit needs one that is longer than 35 seconds and shorter than 70 seconds. It must also use the entire recording in complete 15-second sections, with no cutting, skipping, repeating or added silence.
30 s40 s60 s75 s
Length window
Longer than 35 s
Shorter than 70 s
Whole sections
15 s in each section
Use the entire clip
The cards give the available lengths. Every stated condition applies to the chosen recording.
Your answer
Enter the usable length. Explain which rule rejects each other supplied recording.
These quantities fit. Your exclusions, completeness and original-condition checks remain unscored.
Break it into steps
Step 1 of 3
How many supplied lengths are strictly within the time window?
Forty and sixty lie in the required window.
Step 2 of 3
How many complete fifteen-second sections does the usable recording make?
Sixty seconds makes four complete fifteen-second sections.
Step 3 of 3
Which recording length works?
The sixty-second recording is the sole permitted choice.
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
What does each clue tell you on its own? A usable choice must satisfy all the rules together.
Show the answer and reasoning: the recording
Answer: The 60-second recording works. It makes four complete 15-second sections.
The length window permits 40 and 60 seconds. Complete 15-second sections permit 30, 60 and 75 seconds. Keep only a choice allowed by both clues.
30 seconds
Window: fails
15 s sections: complete
Reject
40 seconds
Window: fits
15 s sections: incomplete
Reject
60 seconds
Window: fits
15 s sections: complete
Fits both
75 seconds
Window: fails
15 s sections: complete
Reject
Each clue permits more than one recording. Their common choice is the sixty-second clip.
Mina “Forty fits the length window. I nearly chose it straight away.”
CKRobo “What does your other check establish?”
Rowan “Forty leaves a piece after two fifteen-second sections. Sixty passes both checks.”
Check: \(60=4\times15\), and 60 is strictly between 35 and 70. Thirty and 75 make complete sections but fail the length window. Forty meets the window but cannot be split into complete 15-second sections under the stated rules.
Use logical reasoning
Use the stated rules and useful properties to decide what must follow. A conclusion needs a reason: a clue it satisfies, a contradiction it avoids, or a connection between facts.
Combine. Which possibilities fit every relevant clue?
Exclude carefully. Which exact rule forbids this choice?
Keep open. Could more than one explanation still fit?
Two weak clues can be powerful together. But combining clues does not always leave one answer. Keep track of what is known, what is ruled out and what remains unresolved.
Start with what you know helps you select a useful first part. Here we ask why a conclusion follows, and how far it really takes us.
Cross off used values
Mina prepares a programme that lets every visitor group hear all three clips. A number used already can rule out another placement—but only where a rule forbids the repeat.
Three listening stations, A, B and C, run three rounds. Each entry in the programme is a clip number: 1, 2 or 3.
In each round, all three clips play once each across the stations. At each station, all three clips play once each across the rounds. So every row and every column contains 1, 2 and 3 once.
The only known entry is Round 1, Station A: clip 1. All other entries are unknown.
Which other cells can already be ruled out for clip 1? Is clip 1 ruled out at Round 2, Station B? If it is possible there, give a complete valid programme showing it.
CKSTEM Thinking CompassPlan and check your reasoning.
Which other cells can already be ruled out for clip 1? Is clip 1 ruled out at Round 2, Station B? If it is possible there, give a complete valid programme showing it.
Cross off used values
Mina prepares a programme that lets every visitor group hear all three clips. A number used already can rule out another placement—but only where a rule forbids the repeat.
Three listening stations, A, B and C, run three rounds. Each entry in the programme is a clip number: 1, 2 or 3.
In each round, all three clips play once each across the stations. At each station, all three clips play once each across the rounds. So every row and every column contains 1, 2 and 3 once.
The only known entry is Round 1, Station A: clip 1. All other entries are unknown.
Known programme entry
Round
A
B
C
1
1
?
?
2
?
?
?
3
?
?
?
Columns are stations A, B and C. Question marks are unresolved entries, not rejected clips.
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.
Known programme entry
Round
A
B
C
1
1
?
?
2
?
?
?
3
?
?
?
Columns are stations A, B and C. Question marks are unresolved entries, not rejected clips.
Show the answer and reasoning: where a repeat is forbidden
Answer: Clip 1 is ruled out at Round 1/B, Round 1/C, Round 2/A and Round 3/A. It is possible at Round 2/B, as the complete programme below shows.
Round 1 already contains 1, so its other two entries cannot be 1. Station A already contains 1, so its later two entries cannot be 1. Those are the four cells sharing a no-repeat rule with the known entry.
What the first 1 rules out
Round
A
B
C
1
1
not 1
not 1
2
not 1
?
?
3
not 1
?
?
“not 1” marks an exclusion of this value only. It does not make the cell unused; clip 2 or 3 must eventually fill it.
Round 2/B shares neither that row nor that column. The first entry does not rule it out. A complete programme is a stronger check that it really can work:
A complete programme with 1 at Round 2/B
Round
A
B
C
1
1
2
3
2
3
1
2
3
2
3
1
Every row and column has each clip once. The outlined entry is compatible with the first known 1.
Rowan “I crossed out every other one. I treated the whole board like a once-only bag of tiles.”
CKRobo “Where does the programme actually forbid a repeat?”
Mina “Within a round or a station. We need three copies of each clip over the whole programme.”
Check: each row and each column of the completed example contains 1, 2 and 3 once. Round 2/B is 1. The known first entry cannot justify crossing it off there. A question mark means “not settled”; it does not mean “forbidden”.
Place the last copy
A separate three-round programme has the same full rules: stations A, B and C each play clips 1, 2 and 3 once, and each round also uses 1, 2 and 3 once.
This time, Round 1/A is 1 and Round 2/C is 1. All other entries are unknown.
Where must the third copy of clip 1 go? Do these clues also force every other entry? If not, show two complete valid programmes.
CKSTEM Thinking CompassPlan and check your reasoning.
Where must the third copy of clip 1 go? Do these clues also force every other entry? If not, show two complete valid programmes.
Place the last copy
A separate three-round programme has the same full rules: stations A, B and C each play clips 1, 2 and 3 once, and each round also uses 1, 2 and 3 once.
This time, Round 1/A is 1 and Round 2/C is 1. All other entries are unknown.
A separate programme
Round
A
B
C
1
1
?
?
2
?
?
1
3
?
?
?
Only the two displayed entries are known. The same row and column rules apply.
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.
A separate programme
Round
A
B
C
1
1
?
?
2
?
?
1
3
?
?
?
Only the two displayed entries are known. The same row and column rules apply.
Show the answer and reasoning: the final copy
Answer: The third 1 must go at Round 3, Station B. The other entries are not all forced; the two programmes below both work.
Rounds 1 and 2 already have their 1. Round 3 still needs one. Station A has its 1 in Round 1, and C has its 1 in Round 2. Only Station B remains for Round 3’s 1.
Complete programme 1
Programme 1
Round
A
B
C
1
1
2
3
2
2
3
1
3
3
1
2
Complete programme 2
Programme 2
Round
A
B
C
1
1
3
2
2
3
2
1
3
2
1
3
Both programmes require the third 1 at Round 3/B. The other numbers can differ, so the whole programme is not uniquely fixed.
Check: every row and column in both examples contains 1, 2 and 3 once. They keep the two original clues and agree on Round 3/B, but disagree about the 2s and 3s. Finding one necessary entry is different from proving that the whole programme is unique.
What does the lamp tell us?
A model listening station plays exactly one of clips 1, 2 and 3. Its blue lamp is on exactly when clip 3 is playing or the manual test button is pressed. “Or” includes the case when both happen. There are no other controls.
The lamp is on, but the test button’s state is unknown.
Can you determine one unique playing clip? If it is forced, identify it and explain why. Otherwise, show two allowed cases with different clips.
CKSTEM Thinking CompassPlan and check your reasoning.
Can you determine one unique playing clip? If it is forced, identify it and explain why. Otherwise, show two allowed cases with different clips.
What does the lamp tell us?
A model listening station plays exactly one of clips 1, 2 and 3. Its blue lamp is on exactly when clip 3 is playing or the manual test button is pressed. “Or” includes the case when both happen. There are no other controls.
The lamp is on, but the test button’s state is unknown.
Observed lampOn
Playing clipUnknown
Exactly one of 1, 2 or 3
Test buttonState unknown
The lamp and the complete stated control rule are the evidence. The clip and button state have not been recorded.
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.
Observed lampOn
Playing clipUnknown
Exactly one of 1, 2 or 3
Test buttonState unknown
The lamp and the complete stated control rule are the evidence. The clip and button state have not been recorded.Show the answer and reasoning: an effect and its causes
Answer:No. Clip 3 with the test button released, or clip 1 with the button pressed, both fit the lit lamp.
Allowed case 1
Clip 3 playing
Test button released
Lamp on
Allowed case 2
Clip 1 playing
Test button pressed
Lamp on
Different playing clips can produce the same observed lamp state under the given rule.
Mina “Clip three always lights the lamp. I thought the lamp proved clip three was playing.”
CKRobo “Do all the allowed explanations lead to the same clip?”
Rowan “No. The test button can light it while clip one or two plays. We have to check more information.”
Check: the first case lights the lamp because clip 3 is playing; the second lights it because the button is pressed. Both satisfy the complete rule. A guaranteed effect does not mean there is only one possible cause.
Recover the recordings
Two recording files have positive whole sizes in megabytes (MB). Harbour’s size is a multiple of 6 MB; Reef’s size is a multiple of 8 MB.
A 6 MB title file is added to the two recordings. For this puzzle’s export model, the entire combined size is then reduced to exactly half, producing a 24 MB export. There are no other files or changes.
Find the original sizes of Harbour and Reef. Explain why no other positive whole pair fits all these clues.
Given export process
? MBTwo recordings
Add 6 MB→↓
? MBWith title
Take half→↓
24 MBExport
The recording sizes are unknown. The title is added before the complete size is halved.Try with the CKSTEM Thinking Compass
Interactive checking needs JavaScript. The complete question and worked answer remain available.
Find the original sizes of Harbour and Reef. Explain why no other positive whole pair fits all these clues.
Recover the recordings
Two recording files have positive whole sizes in megabytes (MB). Harbour’s size is a multiple of 6 MB; Reef’s size is a multiple of 8 MB.
A 6 MB title file is added to the two recordings. For this puzzle’s export model, the entire combined size is then reduced to exactly half, producing a 24 MB export. There are no other files or changes.
Given export process
? MBTwo recordings
Add 6 MB→↓
? MBWith title
Take half→↓
24 MBExport
The recording sizes are unknown. The title is added before the complete size is halved.
Your answer
Enter both original sizes in their labelled roles. Explain why no other pair fits.
These quantities fit. Your exclusions, completeness and original-condition checks remain unscored.
Break it into steps
Step 1 of 4
What combined size existed before the halving?
Twenty-four doubled is forty-eight.
Step 2 of 4
What total did just the two recordings have?
Forty-eight minus six leaves forty-two.
Step 3 of 4
What was Harbour’s original size?
Reef must be a multiple of both six and eight below forty-two: twenty-four. Harbour is eighteen.
Step 4 of 4
What was Reef’s original size?
Twenty-four plus eighteen is forty-two; both size rules fit.
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 two file sizes
Answer:Harbour is 18 MB and Reef is 24 MB. This is the only allowed pair.
Work backwards
First undo the halving, then remove the title file. The two recordings together occupied 42 MB.
Work backwards · recover the earlier totals
24 MBExport
× 2←↓
48 MBBefore halving
− 6 MB←↓
42 MBTwo recordings
Read from the finished export towards the original recordings. Undo halving before removing the title.
\[\begin{aligned}24\times2&=48,\\48-6&=42.\end{aligned}\]
Use logical reasoning to combine the size restrictions. Forty-two is seven groups of six. Harbour takes away complete six-MB groups, so the remaining Reef file must also contain complete six-MB groups. It must therefore be a multiple of both 6 and 8.
Positive multiples of 6 below 42
61218243036
Positive multiples of 8 below 42
816243240
Reef must belong to both lists. Twenty-four is the only common value in the positive range below the combined recording size.
Reef must be smaller than 42 MB because Harbour has a positive size. Only 24 fits both multiple conditions in that range; the next common multiple, 48, is already too large. Harbour is \(42-24=18\) MB.
Check forwards: \(18\div6=3\) and \(24\div8=3\), so both are permitted sizes. The files and title total \(18+24+6=48\) MB; half is the required 24 MB. Positivity rules out an empty recording, and the common-multiple argument covers every allowed pair.
Revisit Work backwards when a final result can reveal an earlier total. That recovered total becomes information for another tool.
A display that uses every tile
For the last exhibit, the programme and tile panel must describe the same mode. The programme clues may leave a choice; a mode is usable only if its complete panel also uses exactly the available tiles.
A four-round programme uses mode codes 1, 2, 3 and 4. Every row and every station column must contain all four codes once each.
The known entries are Round 2/A: 1; Round 2/D: 4; Round 4/C: 4. All other entries are unknown.
The display must use the mode at Round 2, Station C. Its pattern has a central square with a positive whole number of tiles along each side. Below the square are full extra rows of the same width. The number of extra rows is one more than the mode code.
Use all 270 tiles exactly once, with no gaps, overlaps, cuts or other tiles. The extra rows are separate from the central square.
Find the mode that fits both the programme and tile rules, the number of tiles along one side of the central square, and the width and total height of the finished panel. Explain why a different positive whole side count cannot use exactly 270 tiles. You do not need to finish the whole programme.
Mode programme · find Round 2/C
Round
A
B
C
D
1
?
?
?
?
2
1
?
?
4
3
?
?
?
?
4
?
?
4
?
Central square
Same positive whole side count
Extra rows
Number of rows = mode code + 1
Same width as the square
Supply270 tiles
Use every tile once
Columns are stations A–D. The outlined unknown is the requested mode; the other blanks need not all be solved.
Try with the CKSTEM Thinking Compass
Interactive checking needs JavaScript. The complete question and worked answer remain available.
Find the mode that fits both the programme and tile rules, the number of tiles along one side of the central square, and the width and total height of the finished panel. Explain why a different positive whole side count cannot use exactly 270 tiles. You do not need to finish the whole programme.
A display that uses every tile
For the last exhibit, the programme and tile panel must describe the same mode. The programme clues may leave a choice; a mode is usable only if its complete panel also uses exactly the available tiles.
A four-round programme uses mode codes 1, 2, 3 and 4. Every row and every station column must contain all four codes once each.
The known entries are Round 2/A: 1; Round 2/D: 4; Round 4/C: 4. All other entries are unknown.
The display must use the mode at Round 2, Station C. Its pattern has a central square with a positive whole number of tiles along each side. Below the square are full extra rows of the same width. The number of extra rows is one more than the mode code.
Use all 270 tiles exactly once, with no gaps, overlaps, cuts or other tiles. The extra rows are separate from the central square.
Mode programme · find Round 2/C
Round
A
B
C
D
1
?
?
?
?
2
1
?
?
4
3
?
?
?
?
4
?
?
4
?
Central square
Same positive whole side count
Extra rows
Number of rows = mode code + 1
Same width as the square
Supply270 tiles
Use every tile once
Columns are stations A–D. The outlined unknown is the requested mode; the other blanks need not all be solved.
Your answer
Enter the joint mode and dimensions. The numeric checker does not prove programme compatibility, exclusions or uniqueness.
These quantities fit. Your exclusions, completeness and original-condition checks remain unscored.
Break it into steps
Step 1 of 7
What whole side-count ceiling follows because a seventeen-square alone is too large?
Every allowed positive whole side is at most sixteen.
Step 2 of 7
How many tiles does mode 3 need at side fourteen?
Fourteen squared plus four times fourteen gives two hundred and fifty-two.
Step 3 of 7
How many tiles does mode 3 need at side fifteen?
Fifteen squared plus four times fifteen is two hundred and eighty-five. Strict growth leaves no exact whole side for mode three.
Step 4 of 7
Which mode fits both programme and tile rules?
Only mode two can fit an exact positive whole side.
Step 5 of 7
How many tiles are along a central-square side?
Fifteen squared plus forty-five is two hundred and seventy.
Step 6 of 7
How many tiles wide is the finished panel?
The complete panel stays fifteen tiles wide.
Step 7 of 7
How many total rows high is the finished panel?
Fifteen plus three is eighteen.
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 complete display
Answer: Use mode 2. The central square is 15 tiles on each side; the finished panel is 15 tiles wide and 18 rows high. No other positive whole side count works.
Use logical reasoning
Mina “Round 2 has already used one and four, so C must use two or three. Its column excludes four, but it does not yet choose between those two.”
Rowan “Both have complete programmes that work. We must keep both open until we check the same mode against the tile rule.”
CKRobo “What would count as a complete answer, rather than a programme choice that only looks possible?”
Round 2 allows{2, 3}
Station C allows{1, 2, 3}
Both allow{2, 3}
Mode 2 programme
A complete programme with mode two
Round
A
B
C
D
1
2
4
3
1
2
1
3
2
4
3
4
2
1
3
4
3
1
4
2
Mode 3 programme
A complete programme with mode three
Round
A
B
C
D
1
3
4
2
1
2
1
2
3
4
3
4
3
1
2
4
2
1
4
3
Both candidate modes have complete valid programme witnesses. The programme alone does not choose between them; the tile rule must also be met.
A side count \(n\) means \(n^2\) tiles in the square. Mode 2 adds \(3n\) tiles; mode 3 adds \(4n\). We must choose a programme-permitted mode and a positive whole side count that fit together.
Bound the size
If the square had 17 or more tiles along a side, the square alone would need at least \(17^2=289\) tiles. That already exceeds the whole supply. So the positive whole side count is at most 16, before we try any complete panel.
Side count 17 or larger
Central square alone
At least 289 tiles
More than the whole supply
Remaining legal range1 to 16
Positive whole side counts
Every square of side at least seventeen is excluded, without computing the complete panel for each such side.
Guess and check
The programme has left modes 2 and 3. Try a permitted mode and side count, calculate its whole panel, then use “too few” or “too many” to choose a useful next trial. For either mode, a larger positive side count uses more tiles.
Trial 1 · side 12
Square: 144
Extra rows: 36
Total: 180
90 short
Try larger
Trial 2 · side 16
Square: 256
Extra rows: 48
Total: 304
34 too many
Try smaller
Trial 3 · side 14
Square: 196
Extra rows: 42
Total: 238
32 short
Try larger
Trial 4 · side 15
Square: 225
Extra rows: 45
Total: 270
Exact fit
Check the whole plan
Twelve is too small and sixteen too large. Fourteen narrows the remaining whole choice to fifteen; its exact count is then checked.
Rowan “Fourteen uses two hundred and thirty-eight. It is still short, but close.”
CKRobo “What does that tell you about a useful next trial?”
Mina “Try a larger side count. Fifteen gives two hundred and twenty-five in the square and forty-five in the extra rows—exactly two hundred and seventy.”
Mode 2 reaches exactly 270 at side 15. That works, but we still need to decide whether mode 3 gives a second allowed design.
Mode 3 · side 14
14² + 4 · 14 = 252 tiles
Mode 3 · side 15
15² + 4 · 15 = 285 tiles
Mode 3 has four extra rows. Consecutive whole side counts bracket 270 without attaining it; strict growth excludes every other positive whole side.
Rowan “For mode three, side fourteen needs two hundred and fifty-two, but fifteen needs two hundred and eighty-five. There is no whole side between them.”
Mina “The totals grow with the side count. So every smaller side stays below two hundred and seventy and every larger one stays above it. Mode three cannot fit.”
Central square: 225 tiles Three extra rows: 45 tiles
Every outlined tile is counted once. The differently coloured bottom three rows are separate from the fifteen-by-fifteen square.
\[\begin{aligned}15^2+3\times15&=225+45\\&=270,\\15+3&=18.\end{aligned}\]
Why no second side count for mode 2? Increasing a positive side count makes both the square and every added row larger. The total therefore strictly increases. Since 15 gives exactly 270, every smaller value gives less and every larger one gives more.
Check: mode 2 is allowed by the programme constraints and its tile count; a full valid programme is shown with the clue model. The panel has the required three extra rows, no overlap and exactly 270 tiles. Every row is 15 tiles wide, and its height is 18. The complete programme for mode 2 remains a valid witness. Unresolved entries need not be forced to settle the required panel.
How the tools fit: logical clues restrict the possible modes without pretending they are unique; a bound removes impossibly large squares; informative trials test the mode and side together. The tile rule then resolves the programme’s remaining choice. Checking one exact fit and strict growth then proves the side count is unique. Factoring or algebra would also be valid ways to solve the size problem.
Revisit Bound the size when a limit can shrink a search before you try values.
Listening is another kind of observation
A hydrophone is a microphone for listening under water. Scientists use recordings alongside visual observations to learn about marine mammals. A sound can be useful evidence without telling the whole story on its own.
Mina and Rowan’s exhibit uses invented recordings and rules. Their habit is useful beyond the puzzle: listen carefully, say what supports an explanation, and keep room for what is not yet known.
Your turn
Try these before opening the answers. Which conclusions survive the exact rules?
A changed programme rule
Three stations A, B and C again run three rounds with clips 1, 2 and 3. This time, only each round must contain all three clips once. A station may repeat a clip in later rounds; there is no column nonrepetition rule.
The only given entries are Round 1/A: 1 and Round 2/C: 1.
Must Round 3/B still be 1? Justify a forced value, or show two complete programmes that disagree there while meeting these changed rules.
CKSTEM Thinking CompassPlan and check your reasoning.
Must Round 3/B still be 1? Justify a forced value, or show two complete programmes that disagree there while meeting these changed rules.
A changed programme rule
Three stations A, B and C again run three rounds with clips 1, 2 and 3. This time, only each round must contain all three clips once. A station may repeat a clip in later rounds; there is no column nonrepetition rule.
The only given entries are Round 1/A: 1 and Round 2/C: 1.
Changed rule · rows only
Round
A
B
C
1
1
?
?
2
?
?
1
3
?
?
?
Each row uses clips 1, 2 and 3 once. The columns are allowed to repeat under this changed rule.
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.
Changed rule · rows only
Round
A
B
C
1
1
?
?
2
?
?
1
3
?
?
?
Each row uses clips 1, 2 and 3 once. The columns are allowed to repeat under this changed rule.
Show the answer and reasoning: the changed rule
Answer:No. Round 3/B can be 2 in one valid programme and 1 in another.
Allowed programme 1
Rows-only programme 1
Round
A
B
C
1
1
2
3
2
2
3
1
3
1
2
3
Allowed programme 2
Rows-only programme 2
Round
A
B
C
1
1
2
3
2
2
3
1
3
2
1
3
The programmes keep the given entries and disagree at Round 3/B. Each satisfies the rows-only rule.
Check: every row contains 1, 2 and 3 once, and the given entries are kept. Columns may repeat, so neither witness breaks a rule. The earlier last-copy argument depended on the column rule; it cannot be carried over after that rule is removed.
Which booking ID?
A booking ID is one of 24, 30, 36, 42 and 45. It is a multiple of 6 and is strictly between 32 and 44. There are no other clues.
Find every possible ID. Do the clues force one unique ID?
2430364245
These are all the possible IDs before applying the two stated restrictions.Try with the CKSTEM Thinking Compass
Interactive checking needs JavaScript. The complete question and worked answer remain available.
Find every possible ID. Do the clues force one unique ID?
Which booking ID?
A booking ID is one of 24, 30, 36, 42 and 45. It is a multiple of 6 and is strictly between 32 and 44. There are no other clues.
2430364245
These are all the possible IDs before applying the two stated restrictions.
Your answer
List every supplied ID satisfying all clues. Add or remove rows as needed. Decide uniqueness and justify every exclusion separately.
The complete ID set fits. Explain why the clues do not force one unique ID.
Break it into steps
Help step 1
What is the smallest supplied booking ID that meets both clues?
Thirty-six meets the multiple and strict-bound conditions. Keep checking every supplied candidate.
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 booking
Answer:36 and 42 are both possible. No, the ID is not unique.
Twenty-four and 30 are too small; 45 is too large and is not a multiple of 6. The remaining IDs pass both tests: \(36=6\times6\) and \(42=6\times7\). Both are strictly between 32 and 44.
Check: every supplied candidate was considered. Combining clues can leave two possibilities rather than one.
An open model door
In a model, a star ticket opens a door on its own. A circle ticket opens it when a helper’s key is also used. A triangle ticket never opens it. These are the only ways the door can open.
A visitor used one of those three ticket types and the door opened. We do not know whether the helper’s key was used.
Must the ticket have been a star? Could it have been a triangle? Explain every ticket possibility that is consistent with the open door, and why any rejected type is impossible.
CKSTEM Thinking CompassPlan and check your reasoning.
Must the ticket have been a star? Could it have been a triangle? Explain every ticket possibility that is consistent with the open door, and why any rejected type is impossible.
An open model door
In a model, a star ticket opens a door on its own. A circle ticket opens it when a helper’s key is also used. A triangle ticket never opens it. These are the only ways the door can open.
A visitor used one of those three ticket types and the door opened. We do not know whether the helper’s key was used.
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.
Show the answer and reasoning: the door
Answer: A star is not forced; a triangle is impossible. A star on its own or a circle with the helper’s key both explain the open door.
Check: each explanation follows an allowed opening rule, while a triangle contradicts its rule. Knowing that stars open the door does not mean that everything opening it must be a star.
Which clue supports your conclusion?
Ask which clue supports a placement, which rule permits an exclusion and whether another explanation still fits. Do not turn “not known” into “not allowed”, or “this causes that” into “only this can cause that”.
When the rules no longer settle a useful next value, another tool may help. Our next focus is Guess and check: try something meaningful, learn from its result and make the next attempt better.
Rowan “That sound makes me think of a boat. But thinking of something doesn’t prove it.”
CKRobo “What information would let you support your explanation?”
Mina “We can check the recording’s clues. And say what we still don’t know.”