103 practice questions covering Patterns in Mathematics, organised into 1 topics.
103 questions include a written explanation.
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A regular polygon is made from $8$ matchsticks of equal length, one per side. What is the shape called?
AA regular hexagon
BA regular decagon
CA regular heptagon
DA regular octagon
Show answer and explanation
Correct answer: D - A regular octagon
Eight equal sides make a regular octagon. A heptagon has $7$ sides, a hexagon $6$, and a decagon $10$, so none of those could be built from exactly $8$ matchsticks one per side.
Q2
hard
A square hall is tiled so that each new ring of tiles around the centre uses the next odd number of tiles: $1$, then $3$, then $5$, and so on. After the ring of $19$ tiles is laid, how many tiles are in the hall?
A$81$
B$95$
C$100$
D$121$
Show answer and explanation
Correct answer: C - $100$
The odd numbers up to $19$ are the first $10$ odd numbers, and their sum is $10 \times 10 = 100$. Answering $81$ stops at the ring of $17$, $121$ includes the ring of $21$, and $95$ simply adds the numbers incorrectly.
Q3
medium
What is the $9$th triangular number?
A$45$
B$55$
C$81$
D$36$
Show answer and explanation
Correct answer: A - $45$
The $9$th triangular number is $1+2+\cdots+9 = 45$. Answering $36$ gives the $8$th triangular number, $55$ gives the $10$th, and $81$ is the square of $9$ rather than a triangular number.
Q4
medium
Assertion (A): A picture pattern and a number sequence can describe the same growing situation. Reason (R): Every number sequence must come from a picture pattern.
ABoth A and R are true, and R is the correct explanation of A
BBoth A and R are true, but R is not the correct explanation of A
CA is true, but R is false
DA is false, but R is true
Show answer and explanation
Correct answer: C - A is true, but R is false
A is true, since counting the parts of each figure turns a picture pattern into a number sequence. R is false, because many number sequences are defined by a rule alone with no picture behind them. So A is true while R is false.
Q5
medium
What is the sum $1 + 2 + 3 + \cdots + 20$?
A$210$
B$220$
C$190$
D$200$
Show answer and explanation
Correct answer: A - $210$
Pairing $1$ with $20$, $2$ with $19$ and so on gives $10$ pairs each totalling $21$, so the sum is $10 \times 21 = 210$. Answering $200$ or $220$ misjudges the pairing, and $190$ gives the sum up to $19$.
Q6
hard
Which is larger, the $4$th square number or the $3$rd cube number, and by how much?
AThey are equal
BThe cube, by $9$
CThe cube, by $11$
DThe square, by $11$
Show answer and explanation
Correct answer: C - The cube, by $11$
The $4$th square number is $4 \times 4 = 16$ and the $3$rd cube number is $3 \times 3 \times 3 = 27$, so the cube is larger by $27 - 16 = 11$. The square is therefore not the larger of the two, they are not equal, and the gap is $11$ rather than $9$.
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