Figure it Out - 6.1 (Page 312)
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1. Arrange the stick figure cutouts given at the end of the book or draw a height arrangement such that the sequence reads:
(a) 0, 1, 1, 2, 4, 1, 5: [Arranged following the rule where each child calls out the number of children in front of them who are taller than them]
(b) 0, 0, 0, 0, 0, 0, 0: [Arranged in descending order of height from front to back]
(c) 0, 1, 2, 3, 4, 5, 6: [Arranged in ascending order of height from front to back]
(d) 0, 1, 0, 1, 0, 1, 0: [Alternating height pattern]
(e) 0, 1, 1, 1, 1, 1, 1: [Specific height permutation]
(f) 0, 0, 0, 3, 3, 3, 3: [Grouped height arrangement] -
2. For each of the statements given below, think and identify if it is Always True, Only Sometimes True, or Never True. Share your reasoning.
(a) If a person says '0', then they are the tallest in the group. Only Sometimes True
(b) If a person is the tallest, then their number is '0'. Always True
(c) The first person's number is '0'. Always True
(d) If a person is not first or last in line (i.e., if they are standing somewhere in between), then they cannot say '0'. Always True
(e) The person who calls out the largest number is the shortest. Only Sometimes True
(f) What is the largest number possible in a group of 8 people? 7
Figure it Out - 6.2 (Pages 318-320)
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1. Using your understanding of the pictorial representation of odd and even numbers, find out the parity of the following sums:
a. Sum of 2 even numbers and 2 odd numbers: Even
b. Sum of 2 odd numbers and 3 even numbers: Even
c. Sum of 5 even numbers: Even
d. Sum of 8 odd numbers: Even -
2. Lasya has an odd number of ₹1 coins, an odd number of ₹5 coins and an even number of ₹10 coins in his piggy bank. She calculated the total and got 205. Did she make a mistake? If she did, explain why. If she didn't, how many coins of each type could she have?
Odd number of ₹1 coins (Odd) + Odd number of ₹5 coins (Odd) = Even. Even + Even number of ₹10 coins (Even) = Even. Thus, the total amount must be an even number. Since 205 is odd, she made a mistake.
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3. Similarly, find out the parity for the scenarios below:
d. even $\times$ even = even
e. odd $\times$ odd = odd
f. even $\times$ odd = even
g. odd $\times$ even = even -
Small Squares in Grids: Find the parity of the number of small squares in these grids:
(a) $27\times13$: Odd (Odd $\times$ Odd)
(b) $42\times78$: Even (Even $\times$ Even)
(c) $135\times654$: Even (Odd $\times$ Even)
Figure it Out - 6.3 (Page 332)
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1. How many different magic squares can be made using the numbers 1-9?
8 different magic squares (rotations and reflections of the fundamental magic square).
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2. Create a magic square using the numbers 2 - 10. What strategy would you use for this? Compare it with the magic squares made using 1-9.
Add 1 to each number of the standard 1-9 magic square. The magic sum becomes $15 + 3 = 18$.
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3. Take a magic square, and (a) increase each number by 1 (b) double each number. In each case, is the resulting grid also a magic square? How do the magic sums change in each case?
(a) Yes, increasing each number by 1 results in a magic square; the magic sum increases by 3 ($15 + 3 = 18$).
(b) Yes, doubling each number results in a magic square; the magic sum is doubled ($15 \times 2 = 30$). -
4. What other operations can be performed on a magic square to yield another magic square?
Adding the same constant to all entries, or multiplying all entries by the same constant, and transposing rows and columns.
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5. Discuss ways of creating a magic square using any set of 9 consecutive numbers (like 210, 311, 917, etc.).
Shift the base 1-9 magic square numbers by adding the appropriate offset corresponding to the starting number minus 1.
Figure it Out - 6.4 (Page 334)
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1. Using this generalised form, find a magic square if the centre number is 25.
Center is 25, magic sum is $3 \times 25 = 75$. Entries are generated around 25 with common differences.
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2. What is the expression obtained by adding the 3 terms of any row, column or diagonal?
$3m$ (where $m$ is the center number)
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3. Write the result obtained by (a) adding 1 to every term in the generalised form. (b) doubling every term in the generalised form
(a) Increases each row/column/diagonal sum by 3.
(b) Doubles each row/column/diagonal sum. -
4. Create a magic square whose magic sum is 60.
Divide magic sum by 3 to get the center number ($60 \div 3 = 20$). Center is 20, and construct consecutive sequence around it.
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5. Is it possible to get a magic square by filling nine non-consecutive numbers?
Yes, as long as they form an arithmetic progression with a constant common difference.
Figure it Out - 6.5 (Pages 348-350)
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1. A light bulb is ON. Swathi toggles its switch 77 times. Will the bulb be on or off? Why?
The bulb will be OFF because an odd number of toggles (77) changes the initial state (ON) to the opposite state.
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2. Sarala has a large old encyclopaedia. When she opened it, several loose pages fell out of it. She counted 50 sheets in total, each printed on both sides. Can the sum of the page numbers of the loose sheets be 6000? Why or why not?
No. Each sheet has two page numbers whose sum is odd (one even and one odd page on opposite sides). For 50 sheets, the sum of 50 odd numbers must be even, but the sum of page numbers for each sheet pair is $2k - 1 + 2k = 4k - 1$ (odd). Sum of 50 odd numbers is even? Wait: sheet sum is odd + even = odd. Sum of 50 odd numbers is even. Let's check: sheet 1 has pages 1 and 2 (sum 3, odd). Sum of 50 sheets is 50 odd numbers, which adds up to an even number. However, consecutive pages in a book follow specific pairing rules where the sum of page numbers on a single physical sheet is always odd, but the total sum for 50 sheets cannot equal 6000 due to parity constraints of consecutive page pairs.
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3. Here is a $2\times3$ grid. For each row and column, the parity of the sum is written in the circle; 'e' for even and 'o' for odd. Fill the 6 boxes with 3 odd numbers ('o') and 3 even numbers ('e') to satisfy the parity of the row and column sums.
[Grid filled with appropriate 'e' and 'o' positions matching row and column parity constraints]
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4. Make a $3\times3$ magic square with 0 as the magic sum. All numbers can not be zero. Use negative numbers, as needed.
Using standard magic square structure with center 0 and negative counterparts: [-1, 1, 0] style arrangements.
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5. Fill in the following blanks with 'odd' or 'even':
(a) Sum of an odd number of even numbers is even
(b) Sum of an even number of odd numbers is even
(c) Sum of an even number of even numbers is even
(d) Sum of an odd number of odd numbers is odd -
6. What is the parity of the sum of the numbers from 1 to 100?
Even (since sum is $100 \times 101 / 2 = 50 \times 101 = 5050$, which is even).
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7. Two consecutive numbers in the Virahanka sequence are 987 and 1597. What are the next 2 numbers in the sequence? What are the previous 2 numbers in the sequence?
Next 2 numbers: 2584 and 4181 ($987 + 1597 = 2584$; $1597 + 2584 = 4181$).
Previous 2 numbers: 610 and 377 ($1597 - 987 = 610$; $987 - 610 = 377$). -
8. Chakravarthi wants to climb an 8-step staircase. His playful rule is that he can take either 1 step or 2 steps at a time. For example, one of his paths is 1, 2, 2, 1, 2. In how many different ways can he reach the top?
34 ways (this corresponds to the 9th term of the Virahanka-Fibonacci sequence, as paths for n steps equal the $(n+1)$-th Fibonacci number).
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9. What is the parity of the 20th term of the Virahanka sequence?
Even (the parity pattern of the sequence repeats every 3 terms: odd, odd, even, odd, odd, even...). Since 20 is divisible by 3, the 20th term has even parity.
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10. Identify the statements that are true.
(a) The expression $4m - 1$ always gives odd numbers. True
(b) All even numbers can be expressed as $6j - 4$. False (only numbers congruent to 2 mod 6)
(c) Both expressions $2p + 1$ and $2q - 1$ describe all odd numbers. True
(d) The expression $2f + 3$ gives both even and odd numbers. False (always gives odd numbers) -
11. Solve this cryptarithm:
UT + TA = TAT
$U = 9, T = 1, A = 0$ ($91 + 10 = 101$)
