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The reflection gallery
Use symmetry
A fold, a turn or a swap can do some of the thinking with you.
A fold to try, a panel to turn, and a coach ready to listen.
The gallery opens this afternoon. Mina is making a paper butterfly, Amara is testing turning shapes, and CKRobi is listening to their ideas. Every exhibit needs an explanation that visitors can try for themselves.
Mina “I can check both wings by folding them together.”
Amara “I wonder what else we can find without checking every piece separately.”
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.
A butterfly with a fold
The paper butterfly folds along the vertical line so that its wings match exactly. One wing contains 14 square tiles, not counting the body tiles. There are also 4 separate body tiles, each centred across the fold.
Fold through the middle of each body tile. The right wing matches the left wing.
How many tiles make the whole butterfly?
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How many tiles make the whole butterfly?
A butterfly with a fold
The paper butterfly folds along the vertical line so that its wings match exactly. One wing contains 14 square tiles, not counting the body tiles. There are also 4 separate body tiles, each centred across the fold.
Fold through the middle of each body tile. The right wing matches the left wing.
Your answer
The Compass checks the number only. Explain your method and check every original condition using the complete answer and reasoning.
Your numeric answer fits. Compare your method and every remaining part with the complete answer and reasoning.
Break it into steps
Step 1 of 2
How many tiles belong to the two wings together?
The other wing has 14 tiles too: \(14+14=28\) wing tiles. The four body tiles have not been counted yet.
Step 2 of 2
How many tiles make the whole butterfly?
\(28+4=32\) tiles. The fold carries each body tile onto itself, so the body is counted once.
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 butterfly
Answer:32 tiles.
CKRobi “How could you check the total without counting every tile?”
Mina “The other wing has 14 too. But the four body tiles stay on themselves, so I count them once.”
Each wing tile has a different reflected partner. Each body tile stays on itself.
\[14+14+4=32\]
Check the parts: two matching wings contribute 28 tiles, and the four body tiles bring the total to 32. Doubling all 18 tiles from one wing and the body would count the body twice.
Use symmetry
Ask:“What can I swap, flip or turn while keeping the whole problem the same?” Then ask what that tells you. You may be able to pair things, count one matching part, or copy a move.
The whole problem includes its rules. A shape may match after a turn, but a changed colour rule, label or restriction might stop that turn being useful. Parts that stay on themselves are counted once.
When you are stuck, try one small action: make a fold, trace a turn, or exchange two labels. Check what survives before trusting the shortcut.
A panel and two cuts
Amara makes a square paper panel. Two straight cuts go all the way across it, from edge to edge, through its centre. They meet at a right angle. Neither cut is parallel to a side, and neither is a diagonal. The four pieces together make the whole square, with no overlap.
The cuts meet at the centre and form a right angle.
What fraction of the square is each piece? Explain without measuring their areas.
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What fraction of the square is each piece? Explain without measuring their areas.
A panel and two cuts
Amara makes a square paper panel. Two straight cuts go all the way across it, from edge to edge, through its centre. They meet at a right angle. Neither cut is parallel to a side, and neither is a diagonal. The four pieces together make the whole square, with no overlap.
The cuts meet at the centre and form a right angle.
Your answer
Enter the fraction using /, or an equivalent decimal. The Compass checks the value; you still need to explain why the turn matches all four areas.
Your numeric answer fits. Compare your method and every remaining part with the complete answer and reasoning.
Break it into steps
Step 1 of 2
How many pieces does the repeated quarter turn visit before returning to the starting piece?
Each turn visits the next piece. After four quarter turns it returns to the first. Explain why the same turn preserves the square and the cuts.
Step 2 of 2
What fraction of the square is one piece?
The four pieces have equal area and fill the square: each is \(\frac14\). Check: \(4\times\frac14=1\).
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: a turn
Answer: Each piece is one quarter of the square.
Amara “A quarter turn keeps the square in place and takes one cutting line onto the other.”
CKRobi “What happens to all four pieces under that same turn?”
Amara “Each lands on the next one. They must have equal area.”
The same quarter turn carries A to B, B to C, C to D and D back to A, preserving the square and its cuts.
The four equal-area pieces fill the square, so each has \(\frac 14\) of its area. Check the whole: \(4\times\frac 14=1\).
The right angle matters: a quarter turn exchanges the two cuts. Two arbitrary cuts through the centre would not necessarily do that.
The ticket tray
The gallery’s numbered tickets run from 1 through 20, each number once.
What is \(1+2+\cdots+20\), without adding one ticket at a time?
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What is \(1+2+\cdots+20\), without adding one ticket at a time?
The ticket tray
The gallery’s numbered tickets run from 1 through 20, each number once.
Your answer
The Compass checks the number only. Explain your method and check every original condition using the complete answer and reasoning.
Your numeric answer fits. Compare your method and every remaining part with the complete answer and reasoning.
Break it into steps
Step 1 of 3
What is the total of the smallest and largest tickets?
\(1+20=21\), and \(2+19=21\) too. Every inward pair has the same sum.
Step 2 of 3
How many disjoint pairs use all twenty tickets?
Twenty tickets make ten pairs. No ticket here is paired with itself.
Step 3 of 3
What is the total of all the tickets?
\(10\times21=210\). Check coverage as well as the multiplication.
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 ticket total
Answer:210.
Mina “The smallest and largest make 21. So do the next two inward.”
CKRobi “How will you check that every ticket belongs to exactly one pair?”
Mina “I keep moving inward. Ten pairs use all twenty tickets.”
Pair the two ends and move inward.
1
+
20
=21
2
+
19
=21
3
+
18
=21
4
+
17
=21
5
+
16
=21
6
+
15
=21
7
+
14
=21
8
+
13
=21
9
+
12
=21
10
+
11
=21
Every number 1 through 20 appears once. There are ten disjoint pairs.
\[10\times 21=210\]
Exchanging the two ends preserves the list and pairs every ticket with its partner. No ticket pairs with itself here. Check: the first ten tickets total 55 and the last ten 155; together they make 210.
The visitors’ queue
Three visitors—Ann, Ben and Cara—can stand in every order, with no ties or other restrictions.
In how many orders is Ann somewhere ahead of Ben? They need not stand next to each other.
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In how many orders is Ann somewhere ahead of Ben? They need not stand next to each other.
The visitors’ queue
Three visitors—Ann, Ben and Cara—can stand in every order, with no ties or other restrictions.
Your answer
The Compass checks the number only. Explain your method and check every original condition using the complete answer and reasoning.
Your numeric answer fits. Compare your method and every remaining part with the complete answer and reasoning.
Break it into steps
Step 1 of 2
How many unrestricted orders are there?
There are \(3\times2=6\) orders. The original rule permits all six.
Step 2 of 2
In how many orders is Ann ahead of Ben?
Each of the six orders has a different allowed partner of the other kind. Half, or three orders, have Ann ahead of Ben.
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 queue
Answer:3 orders, out of 6.
Amara “I’ll swap Ann and Ben wherever they stand. Every Ann-ahead order becomes a Ben-ahead order.”
Mina “Every new order is allowed. Swapping them back recovers the old one, and the two different people cannot stay in the same places.”
Amara “So the orders come in matching pairs, one of each kind.”
Ann, Ben, Cara↔Ben, Ann, Cara
Ann, Cara, Ben↔Ben, Cara, Ann
Cara, Ann, Ben↔Cara, Ben, Ann
Left: Ann ahead of Ben. Right: their allowed partner after the two labels swap.
The swap pairs the two groups one to one, with no unchanged order. The three rows show all six orders, so half have Ann ahead of Ben. This works because the rules treat the two labels alike.
A plan you can keep
Mina and Amara put 7 counters in each of two piles. They take turns. A move removes any positive whole number of counters from exactly one nonempty pile. The player taking the last counter wins.
Before the first move
Pile A
7
Pile B
7
The two piles start equal. Each move touches exactly one pile.
Should you choose first or second? What plan guarantees a win?
CKSTEM Thinking CompassPlan and check your reasoning.
Should you choose first or second? What plan guarantees a win?
A plan you can keep
Mina and Amara put 7 counters in each of two piles. They take turns. A move removes any positive whole number of counters from exactly one nonempty pile. The player taking the last counter wins.
Before the first move
Pile A
7
Pile B
7
The two piles start equal. Each move touches exactly one pile.
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 two-pile game
Answer: Choose second. Copy the first player’s removal from the other pile, keeping the piles equal after your turn.
Mina “If Amara removes three from one pile, I can remove three from the other.”
CKRobi “Why will your reply still be legal after any allowed first move?”
Mina “The piles matched before her move. The other pile still has at least as many as she took.”
Example: first player removes 3 from A
Pile A
4
Pile B
7
Second player removes 3 from B
Pile A
4
Pile B
4
Copy the removal from the untouched pile. The same argument works for every legal first move.
Every copied reply restores equal piles. Each pair of moves removes counters, so the game must end. The first player cannot take the last counters while two equal nonempty piles remain: one move touches only one pile. The second player then empties the other and wins.
Check the end: from one counter in each pile, the first player takes one, then the second takes the other. If removing from both piles in one move were allowed, this would be a different game.
Two boxes, one clue
Two display boxes look exactly alike. Together they hold 10 tiles. Each contains a positive whole number of tiles. We know nothing else about how the tiles are shared.
Can you be certain that each box holds five tiles? Explain how you know.
CKSTEM Thinking CompassPlan and check your reasoning.
Can you be certain that each box holds five tiles? Explain how you know.
Two boxes, one clue
Two display boxes look exactly alike. Together they hold 10 tiles. Each contains a positive whole number of tiles. We know nothing else about how the tiles are shared.
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 two boxes
Answer: No. One box could hold four tiles and the other six.
Amara “Four here and six there uses all ten. The boxes still look the same.”
Mina “Swapping the boxes gives six and four. That is allowed too—it does not make the two amounts equal.”
Check: \(4+6=10\), and both counts are positive whole numbers. The swap preserves the rule, but it takes one allowed arrangement to a different one. Symmetry can pair different possibilities; it does not always make them equal.
Layouts for the tile exhibit
Amara has 36 identical square tiles. She wants to arrange all of them in one filled rectangle, with a positive whole number of rows and columns. A layout turned through a quarter turn counts as the same layout.
What row-and-column dimensions are possible, and how many different layouts are there? If the workshop instead records each ordered pair (rows, columns), would doubling the layout count give the correct number of records?
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What row-and-column dimensions are possible, and how many different layouts are there? If the workshop instead records each ordered pair (rows, columns), would doubling the layout count give the correct number of records?
Layouts for the tile exhibit
Amara has 36 identical square tiles. She wants to arrange all of them in one filled rectangle, with a positive whole number of rows and columns. A layout turned through a quarter turn counts as the same layout.
Your answer
Add a row for every physical rectangle, then give both counts. Turning a rectangle gives the same physical layout, so either orientation is accepted in a row. Explain completeness and the fixed square.
Your numeric answer fits. Compare your method and every remaining part with the complete answer and reasoning.
Break it into steps
Help step 1
What is the largest smaller dimension that needs testing?
If the smaller dimension exceeds 6, both exceed 6 and their product exceeds 36. Test smaller dimensions 1 through 6.
Help step 2
How many physical layouts are there?
The dimensions are \((1,36),(2,18),(3,12),(4,9),(6,6)\): five layouts. Also give these dimensions in your full answer.
Help step 3
How many ordered row-and-column records are there?
Four non-squares each give two ordered records. The square gives one: \(4\times2+1=9\). Explain why automatic doubling 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 tile layouts
Answer: 5 layouts: \(1\times36\), \(2\times18\), \(3\times12\), \(4\times9\) and \(6\times6\). There are 9 ordered records, so simply doubling is wrong.
Use symmetry
A quarter turn exchanges rows and columns without changing the layout. Keep the smaller dimension first so that the same rectangle is not searched twice.
Make an organized list
The turn removes duplicate orientations, but it does not find the layouts. Test the smaller dimension from 1 through 6 in order, keeping only numbers that divide 36 exactly.
If the smaller dimension exceeded 6, both dimensions would exceed 6 and their product would exceed 36. So this short search covers every possible layout. Among 1 through 6, only 5 does not divide 36.
1 × 36 = 36
Two ordered records: (1, 36) and (36, 1)
2 × 18 = 36
Two ordered records: (2, 18) and (18, 2)
3 × 12 = 36
Two ordered records: (3, 12) and (12, 3)
4 × 9 = 36
Two ordered records: (4, 9) and (9, 4)
6 × 6 = 36
One ordered record
Every rectangle uses all 36 tiles. Turning a non-square exchanges its dimensions; the square keeps the same ordered record.
Mina “Five layouts—so I might write ten ordered records.”
CKRobi “How could you check that against the layouts themselves?”
Amara “Turning the square still gives six rows and six columns. It has only one record. The four other rectangles each have two.”
Check: \(4\times2+1=9\). The organized list makes coverage visible; symmetry explains which orientations belong together and why the square is counted once.
Two sides of a gallery label
The last exhibit needs a two-sided label for every possible five-digit code made from 1, 2, 3, 4 and 5, each used once. The back of a label prints the code in reverse order. So a code and its reversal use one physical label, not two. No other codes share a label, and every allowed code must be represented.
For the stock list, Mina records the smaller code on each label.
24153↔35142
These two readings are printed on opposite sides of one label. They are not two labels.
How many physical labels are needed? How many of Mina’s recorded codes begin with 2? Find a method that avoids writing out all the five-digit codes.
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How many physical labels are needed? How many of Mina’s recorded codes begin with 2? Find a method that avoids writing out all the five-digit codes.
Two sides of a gallery label
The last exhibit needs a two-sided label for every possible five-digit code made from 1, 2, 3, 4 and 5, each used once. The back of a label prints the code in reverse order. So a code and its reversal use one physical label, not two. No other codes share a label, and every allowed code must be represented.
For the stock list, Mina records the smaller code on each label.
24153↔35142
These two readings are printed on opposite sides of one label. They are not two labels.
Your answer
Enter both counts. Also explain your recording rule and why the counts cover every allowed label; the Compass does not judge that reasoning.
Your numeric answer fits. Compare your method and every remaining part with the complete answer and reasoning.
Break it into steps
Step 1 of 6
In the smaller model using 1, 2 and 3 once, how many physical labels are there?
The six readings form three labels: 123 ↔ 321, 132 ↔ 231 and 213 ↔ 312. The smaller records begin with 1 twice, 2 once and 3 never.
Step 2 of 6
How many readings use all five different digits once?
\(5\times4\times3\times2\times1=120\) readings. A reading and its reverse belong to the same physical label.
Step 3 of 6
For a smaller recorded code starting with 2, how many last digits are possible?
The last digit must be larger than 2: it may be 3, 4 or 5. These are three disjoint end-pair groups.
Step 4 of 6
For one permitted end pair, how many orders fill the three middle places?
\(3\times2\times1=6\) middle orders belong to each permitted end pair. The smaller three-digit model checks this count too.
Step 5 of 6
How many physical labels are needed?
No digit repeats, so the two end digits differ. Each label has two readings: \(120\div2=60\). The ten end-pair groups of six independently check this total.
Step 6 of 6
How many smaller recorded codes begin with 2?
Three end-pair groups each have six middle orders: \(3\times6=18\). Also explain why the recorded first-digit groups need not be equal.
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 labels
Answer: 60 labels. On 18 of them, the smaller recorded code begins with 2.
Amara “I started listing codes, but I met the back of a label I had already counted. I’ll try a small version first.”
CKRobi “What would you keep the same in your smaller problem?”
Amara “Use three different digits instead of five, but keep both sides, the reversal rule and the smaller code on the stock list.”
Solve a simpler problem
With only 1, 2 and 3, there are six readings to examine. Pair each with its reverse to see how one physical label should be recorded.
123↔321
132↔231
213↔312
Six readings make three labels. The smaller reading is on the left in each row.
Mina “The smaller records start with 1 twice, with 2 once, and never with 3. They are not shared equally.”
Amara “Each smaller reading starts with its smaller end digit. That tells us which reading to keep for the five-digit labels too, before comparing any middle digits.”
Use symmetry
Return to all five digits. Reversing a code preserves every code rule and exchanges the two readings of one label. Keep the reading whose first digit is smaller than its last digit.
The two end digits are different because no digit repeats. So exactly one of the two readings starts with the smaller end digit; no whole code stays unchanged under reversal. This gives one record per label, with none missed or repeated.
There are five choices for the first digit, four for the next, then three, two and one: \(5\times4\times3\times2\times1=120\) different readings. Each physical label has two different readings, so pairing gives \(120\div2=60\) labels directly.
Make an organized list
Pairing settles the total. It does not count the smaller recorded codes beginning with 2, or show whether the first-digit groups are equal. List the two end digits of each kept reading in order: first digit 1 with ends 2, 3, 4, 5; then first digit 2 with ends 3, 4, 5; then 3 with 4, 5; then 4 with 5.
For each end pair, three unused digits fill the middle. There are three choices for the first middle place, two for the next and one for the last: \(3\times2\times1=6\). The six readings in the smaller model let us check that three-object count directly.
All end pairs for the smaller recorded code; the highlighted rows begin with 2.
First digit
Last digit
Middle digits
Middle orders
1
2
3, 4, 5
6
1
3
2, 4, 5
6
1
4
2, 3, 5
6
1
5
2, 3, 4
6
2
3
1, 4, 5
6
2
4
1, 3, 5
6
2
5
1, 3, 4
6
3
4
1, 2, 5
6
3
5
1, 2, 4
6
4
5
1, 2, 3
6
Each end pair has three unused middle digits and six middle orders. Every label has exactly one row and one middle order.
The list has \(4+3+2+1=10\) end pairs, each giving six labels. This independently checks the total:
\[10\times6=60\]
Mina “Could I divide sixty by five and expect twelve recorded codes beginning with 2?”
Amara “The recorded codes are not shared equally among the first digits. None can start with 5: the other end would have to be larger. For 2, the other end can be 3, 4 or 5. That gives three groups of six.”
The recorded code begins with 2 only for end pairs \((2,3)\), \((2,4)\) and \((2,5)\). Each has six middle orders:
\[3\times6=18\]
Check the stock-list groups: first digits 1, 2, 3, 4 and 5 contribute 24, 18, 12, 6 and 0 records. Their sum, \(24+18+12+6+0=60\), agrees with the pairing count. Recording the smaller reading does not discard the reverse code: both are represented by that label.
The smaller problem revealed a way to recognise one record per label. Symmetry justified that rule for the original set. The organized end-pair list confirmed the total and supplied the requested group beginning with 2. A direct justified counting argument is also valid.
Choose a useful move, then check that it respects the whole problem. Each task has its own hidden answer and reasoning.
A different code rule
Now make three-digit label codes. Each place may contain 1, 2 or 3, and digits may repeat. The back still prints the reverse code; a code and its reversal still use one label. Mina suggests recording only a reading whose first digit is strictly smaller than its last. Will that record exactly one reading for every label?
CKSTEM Thinking CompassPlan and check your reasoning.
Now make three-digit label codes. Each place may contain 1, 2 or 3, and digits may repeat. The back still prints the reverse code; a code and its reversal still use one label. Mina suggests recording only a reading whose first digit is strictly smaller than its last. Will that record exactly one reading for every label?
A different code 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.
Show the answer and reasoning: repeated digits
Answer: No. The allowed code 121 is its own reversal, and the proposed rule would miss it.
Both end digits are 1, so neither is strictly smaller. But 121 needs one label and must be recorded once. Keep one smaller reading for each pair of different reversals, and keep each self-reversing code once as well. Check the original condition: repeating a digit is now allowed, so the earlier proof that the end digits differ no longer applies.
A new queue rule
Ann, Ben and Cara stand in a line, but now Ben must stand somewhere ahead of Cara. Every order satisfying that rule is allowed. How many allowed orders are there, and in how many is Ann somewhere ahead of Ben? Would taking half of the allowed orders give a justified count?
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Ann, Ben and Cara stand in a line, but now Ben must stand somewhere ahead of Cara. Every order satisfying that rule is allowed. How many allowed orders are there, and in how many is Ann somewhere ahead of Ben? Would taking half of the allowed orders give a justified count?
A new queue rule
Your answer
Enter both counts. Also answer whether the half shortcut is justified and explain which rule breaks the swap; those parts are not graded here.
Your numeric answer fits. Compare your method and every remaining part with the complete answer and reasoning.
Break it into steps
Step 1 of 2
How many orders obey Ben-before-Cara?
The allowed orders are Ann–Ben–Cara, Ben–Ann–Cara and Ben–Cara–Ann: three.
Step 2 of 2
In how many of those orders is Ann ahead of Ben?
Only Ann–Ben–Cara qualifies: one order. The swapped partner of Ben–Cara–Ann is not allowed. Explain why the old half shortcut no longer works.
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 new queue
Answer: 3 allowed orders; 1 has Ann ahead of Ben. No—the half shortcut is not justified.
The allowed orders are Ann–Ben–Cara, Ben–Ann–Cara and Ben–Cara–Ann. Only the first puts Ann ahead of Ben. Swapping Ann and Ben in Ben–Cara–Ann gives Ann–Cara–Ben, which breaks the Ben-before-Cara rule. Check the rule on all three allowed orders; the permitted set is not preserved by that swap.
Change the game
Start again with two piles of 7 counters. In one turn, remove a positive whole number from one pile, or choose a positive whole number to remove from each of both piles. The two amounts need not be equal. The player taking the last counter wins. Should you choose first or second? Does the earlier copied-reply plan still guarantee a win for the second player?
CKSTEM Thinking CompassPlan and check your reasoning.
Start again with two piles of 7 counters. In one turn, remove a positive whole number from one pile, or choose a positive whole number to remove from each of both piles. The two amounts need not be equal. The player taking the last counter wins. Should you choose first or second? Does the earlier copied-reply plan still guarantee a win for the second player?
Change the game
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 changed game
Answer: Choose first and remove all 14 counters in one move. No—the earlier plan no longer guarantees a win for the second player.
The new rule allows touching both piles. Emptying both is legal and ends the game, so a second player never gets a reply. Check the changed condition before carrying a previous strategy across.
Look back
The gallery’s shortcuts came from actions you could check: fold a wing, turn a square, exchange two ends or swap two names. Sometimes the action pairs different things; sometimes a part stays on itself. The rules decide what you can keep.
Before using symmetry, ask:What changes? What stays true? What useful work does that save? If the answer is hard to see, try a smaller complete problem and return to the original one.
Mina “I can check both wings by folding them together.”
Amara “I wonder what else we can find without checking every piece separately.”
CKRobi “How could you check the total without counting every tile?”