Focus and Screen Time · 27 original mini-challenges
Short Reasoning Rounds
27 independent pattern, recall and decision questions with explained answers. Nine families contain three cases each.
Choose a response. See the reasoning.
Read the facts, pick the answer that best fits the stated goal, then open the explanation. Start with five or browse the full set. There is no timer or personal assessment; the useful part is understanding the choice.
These are original web companion exercises. They are independent of the app; answers and progress do not transfer. All people and records in the scenarios are fictional. These puzzles do not measure brain age, intelligence or a health condition.
01 / reasoning
Growing gaps case 1
Continue the sequence 2, 5, 9, 14. The rule is that each gap between consecutive numbers is exactly one larger than the previous gap. Apply that rule once more. Which number must come immediately after the final displayed value?
Show the worked answer
A. 20
The displayed gaps are 3, 4, 5, so the next gap is 6. Adding it to 14 gives 20. The explicitly stated gap rule determines this continuation.
Try next: Write one further value and label the gap used.
Growing gaps case 1
Continue the sequence 2, 5, 9, 14. The rule is that each gap between consecutive numbers is exactly one larger than the previous gap. Apply that rule once more. Which number must come immediately after the final displayed value?
A. 20
B. 19
C. 21
Suggested answer: A. 20
The displayed gaps are 3, 4, 5, so the next gap is 6. Adding it to 14 gives 20. The explicitly stated gap rule determines this continuation.
Next: Write one further value and label the gap used.
Original web companion by Honest Little Apps. Explore SkillArena: Focus Training: https://apps.apple.com/app/id679571330702 / reasoning
Parity without full addition case 1
A score is the sum of 3, 8, 11, 6. Only its parity matters: even means divisible by two, and odd means not divisible by two. Without needing the exact total, which parity must the score have?
Show the worked answer
B. Even
There are 2 odd addends. Odd addends pair into even sums, while every even addend preserves parity. The resulting total is 28, confirming that the score is even.
Try next: Replace one even addend with the next integer and predict the new parity.
Parity without full addition case 1
A score is the sum of 3, 8, 11, 6. Only its parity matters: even means divisible by two, and odd means not divisible by two. Without needing the exact total, which parity must the score have?
A. Odd
B. Even
C. Cannot be determined
Suggested answer: B. Even
There are 2 odd addends. Odd addends pair into even sums, while every even addend preserves parity. The resulting total is 28, confirming that the score is even.
Next: Replace one even addend with the next integer and predict the new parity.
Original web companion by Honest Little Apps. Explore SkillArena: Focus Training: https://apps.apple.com/app/id679571330703 / reasoning
Rank the scattered values case 1
Five different values are 8, 3, 11, 6, 9. Arrange them from largest to smallest, counting the largest as position one. Which value occupies position 2? Each printed value appears once, and no tie-breaking rule is needed.
Show the worked answer
C. 9
Descending order is 11, 9, 8, 6, 3. Counting from the largest places 9 at position 2. Sorting before selecting a position prevents confusing a value’s printed location with its numerical rank.
Try next: Identify the same value’s position when the order is reversed.
Rank the scattered values case 1
Five different values are 8, 3, 11, 6, 9. Arrange them from largest to smallest, counting the largest as position one. Which value occupies position 2? Each printed value appears once, and no tie-breaking rule is needed.
A. 11
B. 3
C. 9
Suggested answer: C. 9
Descending order is 11, 9, 8, 6, 3. Counting from the largest places 9 at position 2. Sorting before selecting a position prevents confusing a value’s printed location with its numerical rank.
Next: Identify the same value’s position when the order is reversed.
Original web companion by Honest Little Apps. Explore SkillArena: Focus Training: https://apps.apple.com/app/id679571330704 / reasoning
Rotate the row case 1
The row is A, B, C, D. Rotate it right by 1 position: each right rotation moves the last item to the front and shifts the others right. Which row results after all rotations, with no items added or removed?
Show the worked answer
A. D A B C
The resulting row is D A B C. Move the ending block to the front while preserving its internal order; a rotation changes positions but does not reverse the row or the moved block.
Try next: Undo the move using the same number of left rotations.
Rotate the row case 1
The row is A, B, C, D. Rotate it right by 1 position: each right rotation moves the last item to the front and shifts the others right. Which row results after all rotations, with no items added or removed?
A. D A B C
B. A B C D
C. B C D A
Suggested answer: A. D A B C
The resulting row is D A B C. Move the ending block to the front while preserving its internal order; a rotation changes positions but does not reverse the row or the moved block.
Next: Undo the move using the same number of left rotations.
Original web companion by Honest Little Apps. Explore SkillArena: Focus Training: https://apps.apple.com/app/id679571330705 / reasoning
Two-condition gate case 1
A number gate opens only when a value is greater than 5 and divisible by 2. Both conditions are required together. The presented value is 8. Does the gate open, or does at least one required condition fail?
Show the worked answer
B. Opens
8 is greater than 5, and 8 is divisible by 2. Therefore the gate opens. Passing only one condition would never satisfy an AND rule.
Try next: Find the smallest positive whole number that would open this gate.
Two-condition gate case 1
A number gate opens only when a value is greater than 5 and divisible by 2. Both conditions are required together. The presented value is 8. Does the gate open, or does at least one required condition fail?
A. Stays closed
B. Opens
C. Insufficient information
Suggested answer: B. Opens
8 is greater than 5, and 8 is divisible by 2. Therefore the gate opens. Passing only one condition would never satisfy an AND rule.
Next: Find the smallest positive whole number that would open this gate.
Original web companion by Honest Little Apps. Explore SkillArena: Focus Training: https://apps.apple.com/app/id679571330706 / reasoning
Fixed first tile case 1
Three different tiles A, B and C must form a row, using every tile exactly once. The first tile is required to be A. How many different complete rows remain possible? A row and its reversal count separately when both are allowed.
Show the worked answer
C. 2
Fixing A leaves two unused tiles and two spaces. Either remaining tile can occupy the second space, after which the last is forced. Therefore exactly two complete rows satisfy the first-position restriction.
Try next: Write both valid rows and verify that neither repeats a tile.
Fixed first tile case 1
Three different tiles A, B and C must form a row, using every tile exactly once. The first tile is required to be A. How many different complete rows remain possible? A row and its reversal count separately when both are allowed.
A. 1
B. 6
C. 2
Suggested answer: C. 2
Fixing A leaves two unused tiles and two spaces. Either remaining tile can occupy the second space, after which the last is forced. Therefore exactly two complete rows satisfy the first-position restriction.
Next: Write both valid rows and verify that neither repeats a tile.
Original web companion by Honest Little Apps. Explore SkillArena: Focus Training: https://apps.apple.com/app/id679571330707 / reasoning
Negate the comparison case 1
Consider the claim “the number 4 is not greater than 6.” Interpret “not greater than” as less than or equal to, including equality. Is the claim true or false under the ordinary ordering of whole numbers?
Show the worked answer
A. True
4 is less than or equal to 6, so the claim is true. Negating “greater than” includes the equality case; it does not simply mean “less than.”
Try next: Rewrite the same claim using a mathematical inequality.
Negate the comparison case 1
Consider the claim “the number 4 is not greater than 6.” Interpret “not greater than” as less than or equal to, including equality. Is the claim true or false under the ordinary ordering of whole numbers?
A. True
B. False
C. Cannot be checked
Suggested answer: A. True
4 is less than or equal to 6, so the claim is true. Negating “greater than” includes the equality case; it does not simply mean “less than.”
Next: Rewrite the same claim using a mathematical inequality.
Original web companion by Honest Little Apps. Explore SkillArena: Focus Training: https://apps.apple.com/app/id679571330708 / reasoning
Insert before recalling case 1
Start with the ordered list oak, elm, ash, fir. Insert pine immediately after item 1, counting from one, and shift later items right without deleting anything. Which item occupies position 3 in the resulting five-item list?
Show the worked answer
B. elm
The new list is oak, pine, elm, ash, fir. Position 3 holds elm. Inserting an item preserves the original relative order, but every item after the insertion point moves one position right.
Try next: Name the item that remains in its original position at the front.
Insert before recalling case 1
Start with the ordered list oak, elm, ash, fir. Insert pine immediately after item 1, counting from one, and shift later items right without deleting anything. Which item occupies position 3 in the resulting five-item list?
A. pine
B. elm
C. oak
Suggested answer: B. elm
The new list is oak, pine, elm, ash, fir. Position 3 holds elm. Inserting an item preserves the original relative order, but every item after the insertion point moves one position right.
Next: Name the item that remains in its original position at the front.
Original web companion by Honest Little Apps. Explore SkillArena: Focus Training: https://apps.apple.com/app/id679571330709 / reasoning
Quarter-turn compass case 1
A marker faces north. Apply these quarter-turns in order: right, right, left. Each right turn is clockwise by 90 degrees; each left turn is counterclockwise by 90 degrees. The marker stays in place. Which direction does it finally face?
Show the worked answer
C. east
The final direction is east. Opposite left and right quarter-turns cancel, while the remaining turns determine orientation. Turning changes which direction the marker faces without changing where the marker stands.
Try next: Trace every intermediate direction to confirm the final orientation.
Quarter-turn compass case 1
A marker faces north. Apply these quarter-turns in order: right, right, left. Each right turn is clockwise by 90 degrees; each left turn is counterclockwise by 90 degrees. The marker stays in place. Which direction does it finally face?
A. north
B. south
C. east
Suggested answer: C. east
The final direction is east. Opposite left and right quarter-turns cancel, while the remaining turns determine orientation. Turning changes which direction the marker faces without changing where the marker stands.
Next: Trace every intermediate direction to confirm the final orientation.
Original web companion by Honest Little Apps. Explore SkillArena: Focus Training: https://apps.apple.com/app/id679571330710 / reasoning
Growing gaps case 2
Continue the sequence 1, 5, 10, 16. The rule is that each gap between consecutive numbers is exactly one larger than the previous gap. Apply that rule once more. Which number must come immediately after the final displayed value?
Show the worked answer
A. 23
The displayed gaps are 4, 5, 6, so the next gap is 7. Adding it to 16 gives 23. The explicitly stated gap rule determines this continuation.
Try next: Write one further value and label the gap used.
Growing gaps case 2
Continue the sequence 1, 5, 10, 16. The rule is that each gap between consecutive numbers is exactly one larger than the previous gap. Apply that rule once more. Which number must come immediately after the final displayed value?
A. 23
B. 22
C. 24
Suggested answer: A. 23
The displayed gaps are 4, 5, 6, so the next gap is 7. Adding it to 16 gives 23. The explicitly stated gap rule determines this continuation.
Next: Write one further value and label the gap used.
Original web companion by Honest Little Apps. Explore SkillArena: Focus Training: https://apps.apple.com/app/id679571330711 / reasoning
Parity without full addition case 2
A score is the sum of 5, 7, 12, 4. Only its parity matters: even means divisible by two, and odd means not divisible by two. Without needing the exact total, which parity must the score have?
Show the worked answer
B. Even
There are 2 odd addends. Odd addends pair into even sums, while every even addend preserves parity. The resulting total is 28, confirming that the score is even.
Try next: Replace one even addend with the next integer and predict the new parity.
Parity without full addition case 2
A score is the sum of 5, 7, 12, 4. Only its parity matters: even means divisible by two, and odd means not divisible by two. Without needing the exact total, which parity must the score have?
A. Odd
B. Even
C. Cannot be determined
Suggested answer: B. Even
There are 2 odd addends. Odd addends pair into even sums, while every even addend preserves parity. The resulting total is 28, confirming that the score is even.
Next: Replace one even addend with the next integer and predict the new parity.
Original web companion by Honest Little Apps. Explore SkillArena: Focus Training: https://apps.apple.com/app/id679571330712 / reasoning
Rank the scattered values case 2
Five different values are 14, 7, 12, 5, 10. Arrange them from largest to smallest, counting the largest as position one. Which value occupies position 3? Each printed value appears once, and no tie-breaking rule is needed.
Show the worked answer
C. 10
Descending order is 14, 12, 10, 7, 5. Counting from the largest places 10 at position 3. Sorting before selecting a position prevents confusing a value’s printed location with its numerical rank.
Try next: Identify the same value’s position when the order is reversed.
Rank the scattered values case 2
Five different values are 14, 7, 12, 5, 10. Arrange them from largest to smallest, counting the largest as position one. Which value occupies position 3? Each printed value appears once, and no tie-breaking rule is needed.
A. 14
B. 5
C. 10
Suggested answer: C. 10
Descending order is 14, 12, 10, 7, 5. Counting from the largest places 10 at position 3. Sorting before selecting a position prevents confusing a value’s printed location with its numerical rank.
Next: Identify the same value’s position when the order is reversed.
Original web companion by Honest Little Apps. Explore SkillArena: Focus Training: https://apps.apple.com/app/id679571330713 / reasoning
Rotate the row case 2
The row is P, Q, R, S, T. Rotate it right by 2 positions: each right rotation moves the last item to the front and shifts the others right. Which row results after all rotations, with no items added or removed?
Show the worked answer
A. S T P Q R
The resulting row is S T P Q R. Move the ending block to the front while preserving its internal order; a rotation changes positions but does not reverse the row or the moved block.
Try next: Undo the move using the same number of left rotations.
Rotate the row case 2
The row is P, Q, R, S, T. Rotate it right by 2 positions: each right rotation moves the last item to the front and shifts the others right. Which row results after all rotations, with no items added or removed?
A. S T P Q R
B. P Q R S T
C. R S T P Q
Suggested answer: A. S T P Q R
The resulting row is S T P Q R. Move the ending block to the front while preserving its internal order; a rotation changes positions but does not reverse the row or the moved block.
Next: Undo the move using the same number of left rotations.
Original web companion by Honest Little Apps. Explore SkillArena: Focus Training: https://apps.apple.com/app/id679571330714 / reasoning
Two-condition gate case 2
A number gate opens only when a value is greater than 8 and divisible by 3. Both conditions are required together. The presented value is 11. Does the gate open, or does at least one required condition fail?
Show the worked answer
B. Stays closed
11 is greater than 8, and 11 is not divisible by 3. Therefore the gate stays closed. Passing only one condition would never satisfy an AND rule.
Try next: Find the smallest positive whole number that would open this gate.
Two-condition gate case 2
A number gate opens only when a value is greater than 8 and divisible by 3. Both conditions are required together. The presented value is 11. Does the gate open, or does at least one required condition fail?
A. Opens
B. Stays closed
C. Insufficient information
Suggested answer: B. Stays closed
11 is greater than 8, and 11 is not divisible by 3. Therefore the gate stays closed. Passing only one condition would never satisfy an AND rule.
Next: Find the smallest positive whole number that would open this gate.
Original web companion by Honest Little Apps. Explore SkillArena: Focus Training: https://apps.apple.com/app/id679571330715 / reasoning
Fixed first tile case 2
Three different tiles A, B and C form a row, using every tile exactly once. B is forbidden from the first position but may appear second or third. How many different complete rows satisfy that restriction?
Show the worked answer
B. 4
There are two choices for the first tile, A or C. For either choice, the remaining two tiles can be arranged in two ways. That gives four valid rows, rather than the two possible with a fixed first tile.
Try next: List the rows and cross out the two beginning with B.
Fixed first tile case 2
Three different tiles A, B and C form a row, using every tile exactly once. B is forbidden from the first position but may appear second or third. How many different complete rows satisfy that restriction?
A. 2
B. 4
C. 6
Suggested answer: B. 4
There are two choices for the first tile, A or C. For either choice, the remaining two tiles can be arranged in two ways. That gives four valid rows, rather than the two possible with a fixed first tile.
Next: List the rows and cross out the two beginning with B.
Original web companion by Honest Little Apps. Explore SkillArena: Focus Training: https://apps.apple.com/app/id679571330716 / reasoning
Negate the comparison case 2
Consider the claim “the number 7 is not greater than 7.” Interpret “not greater than” as less than or equal to, including equality. Is the claim true or false under the ordinary ordering of whole numbers?
Show the worked answer
A. True
7 is less than or equal to 7, so the claim is true. Negating “greater than” includes the equality case; it does not simply mean “less than.”
Try next: Rewrite the same claim using a mathematical inequality.
Negate the comparison case 2
Consider the claim “the number 7 is not greater than 7.” Interpret “not greater than” as less than or equal to, including equality. Is the claim true or false under the ordinary ordering of whole numbers?
A. True
B. False
C. Cannot be checked
Suggested answer: A. True
7 is less than or equal to 7, so the claim is true. Negating “greater than” includes the equality case; it does not simply mean “less than.”
Next: Rewrite the same claim using a mathematical inequality.
Original web companion by Honest Little Apps. Explore SkillArena: Focus Training: https://apps.apple.com/app/id679571330717 / reasoning
Insert before recalling case 2
Start with the ordered list red, blue, gold, green. Insert white immediately after item 2, counting from one, and shift later items right without deleting anything. Which item occupies position 4 in the resulting five-item list?
Show the worked answer
B. gold
The new list is red, blue, white, gold, green. Position 4 holds gold. Inserting an item preserves the original relative order, but every item after the insertion point moves one position right.
Try next: Name the item that remains in its original position at the front.
Insert before recalling case 2
Start with the ordered list red, blue, gold, green. Insert white immediately after item 2, counting from one, and shift later items right without deleting anything. Which item occupies position 4 in the resulting five-item list?
A. white
B. gold
C. red
Suggested answer: B. gold
The new list is red, blue, white, gold, green. Position 4 holds gold. Inserting an item preserves the original relative order, but every item after the insertion point moves one position right.
Next: Name the item that remains in its original position at the front.
Original web companion by Honest Little Apps. Explore SkillArena: Focus Training: https://apps.apple.com/app/id679571330718 / reasoning
Quarter-turn compass case 2
A marker faces east. Apply these quarter-turns in order: left, left, right. Each right turn is clockwise by 90 degrees; each left turn is counterclockwise by 90 degrees. The marker stays in place. Which direction does it finally face?
Show the worked answer
C. north
The final direction is north. Opposite left and right quarter-turns cancel, while the remaining turns determine orientation. Turning changes which direction the marker faces without changing where the marker stands.
Try next: Trace every intermediate direction to confirm the final orientation.
Quarter-turn compass case 2
A marker faces east. Apply these quarter-turns in order: left, left, right. Each right turn is clockwise by 90 degrees; each left turn is counterclockwise by 90 degrees. The marker stays in place. Which direction does it finally face?
A. east
B. south
C. north
Suggested answer: C. north
The final direction is north. Opposite left and right quarter-turns cancel, while the remaining turns determine orientation. Turning changes which direction the marker faces without changing where the marker stands.
Next: Trace every intermediate direction to confirm the final orientation.
Original web companion by Honest Little Apps. Explore SkillArena: Focus Training: https://apps.apple.com/app/id679571330719 / reasoning
Growing gaps case 3
Continue the sequence 4, 6, 9, 13. The rule is that each gap between consecutive numbers is exactly one larger than the previous gap. Apply that rule once more. Which number must come immediately after the final displayed value?
Show the worked answer
A. 18
The displayed gaps are 2, 3, 4, so the next gap is 5. Adding it to 13 gives 18. The explicitly stated gap rule determines this continuation.
Try next: Write one further value and label the gap used.
Growing gaps case 3
Continue the sequence 4, 6, 9, 13. The rule is that each gap between consecutive numbers is exactly one larger than the previous gap. Apply that rule once more. Which number must come immediately after the final displayed value?
A. 18
B. 17
C. 19
Suggested answer: A. 18
The displayed gaps are 2, 3, 4, so the next gap is 5. Adding it to 13 gives 18. The explicitly stated gap rule determines this continuation.
Next: Write one further value and label the gap used.
Original web companion by Honest Little Apps. Explore SkillArena: Focus Training: https://apps.apple.com/app/id679571330720 / reasoning
Parity without full addition case 3
A score is the sum of 2, 9, 14, 7. Only its parity matters: even means divisible by two, and odd means not divisible by two. Without needing the exact total, which parity must the score have?
Show the worked answer
B. Even
There are 2 odd addends. Odd addends pair into even sums, while every even addend preserves parity. The resulting total is 32, confirming that the score is even.
Try next: Replace one even addend with the next integer and predict the new parity.
Parity without full addition case 3
A score is the sum of 2, 9, 14, 7. Only its parity matters: even means divisible by two, and odd means not divisible by two. Without needing the exact total, which parity must the score have?
A. Odd
B. Even
C. Cannot be determined
Suggested answer: B. Even
There are 2 odd addends. Odd addends pair into even sums, while every even addend preserves parity. The resulting total is 32, confirming that the score is even.
Next: Replace one even addend with the next integer and predict the new parity.
Original web companion by Honest Little Apps. Explore SkillArena: Focus Training: https://apps.apple.com/app/id679571330721 / reasoning
Rank the scattered values case 3
Five different values are 19, 21, 15, 18, 17. Arrange them from largest to smallest, counting the largest as position one. Which value occupies position 4? Each printed value appears once, and no tie-breaking rule is needed.
Show the worked answer
C. 17
Descending order is 21, 19, 18, 17, 15. Counting from the largest places 17 at position 4. Sorting before selecting a position prevents confusing a value’s printed location with its numerical rank.
Try next: Identify the same value’s position when the order is reversed.
Rank the scattered values case 3
Five different values are 19, 21, 15, 18, 17. Arrange them from largest to smallest, counting the largest as position one. Which value occupies position 4? Each printed value appears once, and no tie-breaking rule is needed.
A. 21
B. 15
C. 17
Suggested answer: C. 17
Descending order is 21, 19, 18, 17, 15. Counting from the largest places 17 at position 4. Sorting before selecting a position prevents confusing a value’s printed location with its numerical rank.
Next: Identify the same value’s position when the order is reversed.
Original web companion by Honest Little Apps. Explore SkillArena: Focus Training: https://apps.apple.com/app/id679571330722 / reasoning
Rotate the row case 3
The row is L, M, N, O, P, Q. Rotate it right by 3 positions: each right rotation moves the last item to the front and shifts the others right. Which row results after all rotations, with no items added or removed?
Show the worked answer
B. O P Q L M N
The resulting row is O P Q L M N. Move the ending block to the front while preserving its internal order; a rotation changes positions but does not reverse the row or the moved block.
Try next: Undo the move using the same number of left rotations.
Rotate the row case 3
The row is L, M, N, O, P, Q. Rotate it right by 3 positions: each right rotation moves the last item to the front and shifts the others right. Which row results after all rotations, with no items added or removed?
A. L M N O P Q
B. O P Q L M N
Suggested answer: B. O P Q L M N
The resulting row is O P Q L M N. Move the ending block to the front while preserving its internal order; a rotation changes positions but does not reverse the row or the moved block.
Next: Undo the move using the same number of left rotations.
Original web companion by Honest Little Apps. Explore SkillArena: Focus Training: https://apps.apple.com/app/id679571330723 / reasoning
Two-condition gate case 3
A number gate opens only when a value is greater than 10 and divisible by 7. Both conditions are required together. The presented value is 14. Does the gate open, or does at least one required condition fail?
Show the worked answer
B. Opens
14 is greater than 10, and 14 is divisible by 7. Therefore the gate opens. Passing only one condition would never satisfy an AND rule.
Try next: Find the smallest positive whole number that would open this gate.
Two-condition gate case 3
A number gate opens only when a value is greater than 10 and divisible by 7. Both conditions are required together. The presented value is 14. Does the gate open, or does at least one required condition fail?
A. Stays closed
B. Opens
C. Insufficient information
Suggested answer: B. Opens
14 is greater than 10, and 14 is divisible by 7. Therefore the gate opens. Passing only one condition would never satisfy an AND rule.
Next: Find the smallest positive whole number that would open this gate.
Original web companion by Honest Little Apps. Explore SkillArena: Focus Training: https://apps.apple.com/app/id679571330724 / reasoning
Fixed first tile case 3
Three different tiles A, B and C must form a row, each used exactly once. C is fixed first, and A is fixed second. No other tiles may be added. How many complete rows satisfy both fixed-position requirements?
Show the worked answer
A. 1
Only one row works: C, A, B. Fixing the first two positions leaves a single unused tile and a single empty position. The final tile is forced, so there is no remaining ordering choice.
Try next: Remove the second-position requirement and recount the possible rows.
Fixed first tile case 3
Three different tiles A, B and C must form a row, each used exactly once. C is fixed first, and A is fixed second. No other tiles may be added. How many complete rows satisfy both fixed-position requirements?
A. 1
B. 2
C. 3
Suggested answer: A. 1
Only one row works: C, A, B. Fixing the first two positions leaves a single unused tile and a single empty position. The final tile is forced, so there is no remaining ordering choice.
Next: Remove the second-position requirement and recount the possible rows.
Original web companion by Honest Little Apps. Explore SkillArena: Focus Training: https://apps.apple.com/app/id679571330725 / reasoning
Negate the comparison case 3
Consider the claim “the number 10 is not greater than 8.” Interpret “not greater than” as less than or equal to, including equality. Is the claim true or false under the ordinary ordering of whole numbers?
Show the worked answer
A. False
10 is not less than or equal to 8, so the claim is false. Negating “greater than” includes the equality case; it does not simply mean “less than.”
Try next: Rewrite the same claim using a mathematical inequality.
Negate the comparison case 3
Consider the claim “the number 10 is not greater than 8.” Interpret “not greater than” as less than or equal to, including equality. Is the claim true or false under the ordinary ordering of whole numbers?
A. False
B. True
C. Cannot be checked
Suggested answer: A. False
10 is not less than or equal to 8, so the claim is false. Negating “greater than” includes the equality case; it does not simply mean “less than.”
Next: Rewrite the same claim using a mathematical inequality.
Original web companion by Honest Little Apps. Explore SkillArena: Focus Training: https://apps.apple.com/app/id679571330726 / reasoning
Insert before recalling case 3
Start with the ordered list cup, book, lamp, key. Insert bell immediately after item 3, counting from one, and shift later items right without deleting anything. Which item occupies position 5 in the resulting five-item list?
Show the worked answer
B. key
The new list is cup, book, lamp, bell, key. Position 5 holds key. Inserting an item preserves the original relative order, but every item after the insertion point moves one position right.
Try next: Name the item that remains in its original position at the front.
Insert before recalling case 3
Start with the ordered list cup, book, lamp, key. Insert bell immediately after item 3, counting from one, and shift later items right without deleting anything. Which item occupies position 5 in the resulting five-item list?
A. bell
B. key
C. cup
Suggested answer: B. key
The new list is cup, book, lamp, bell, key. Position 5 holds key. Inserting an item preserves the original relative order, but every item after the insertion point moves one position right.
Next: Name the item that remains in its original position at the front.
Original web companion by Honest Little Apps. Explore SkillArena: Focus Training: https://apps.apple.com/app/id679571330727 / reasoning
Quarter-turn compass case 3
A marker faces south. Apply these quarter-turns in order: right, left, left, left. Each right turn is clockwise by 90 degrees; each left turn is counterclockwise by 90 degrees. The marker stays in place. Which direction does it finally face?
Show the worked answer
C. north
The final direction is north. Opposite left and right quarter-turns cancel, while the remaining turns determine orientation. Turning changes which direction the marker faces without changing where the marker stands.
Try next: Trace every intermediate direction to confirm the final orientation.
Quarter-turn compass case 3
A marker faces south. Apply these quarter-turns in order: right, left, left, left. Each right turn is clockwise by 90 degrees; each left turn is counterclockwise by 90 degrees. The marker stays in place. Which direction does it finally face?
A. east
B. south
C. north
Suggested answer: C. north
The final direction is north. Opposite left and right quarter-turns cancel, while the remaining turns determine orientation. Turning changes which direction the marker faces without changing where the marker stands.
Next: Trace every intermediate direction to confirm the final orientation.
Original web companion by Honest Little Apps. Explore SkillArena: Focus Training: https://apps.apple.com/app/id6795713307