Number Play | CBSE Class 7 Maths Notes

This note covers numbers describing height arrangements, odd and even numbers, parity of sums and products, number expressions, row and column sums, magic squares, the Virahāṅka-Fibonacci sequence, counting rhythms, and cryptarithms.

How can numbers describe a height arrangement?

A sequence is an ordered list of numbers. Each number in it is a term. Numbers can describe how people are arranged even when their actual heights are unknown. The important information is their relative height and their position in the line.

In the height activity, each child calls out the number of children standing in front who are taller than that child. Count taller children in front, rather than everyone in front or taller children elsewhere in the group.

What does calling out zero mean?

The first person says 0 because nobody stands in front. The tallest person also says 0 because nobody is taller. However, saying 0 does not establish that a person is the tallest in the whole group.

A child may have nobody taller ahead while a taller child stands behind. This distinction separates a statement that is always true from one that is only sometimes true. A person standing between others can also say 0.

Worked example 1. Under the taller-children-in-front rule, what is the largest possible number called out in a group of 8 people?

Answer: The largest possible number is 7. A person can have at most 7 people ahead. Place the shortest person last to make all 7 people in front taller.

How does changing the order change the sequence?

For seven children of different heights, the sequence 0, 0, 0, 0, 0, 0, 0 occurs when heights increase from front to back. Nobody has a taller child ahead. Reversing that height order gives 0, 1, 2, 3, 4, 5, 6.

In the reversed arrangement, every preceding child is taller. The person calling out the largest number is only sometimes the shortest in a general arrangement, because a shorter child could stand nearer the front and have fewer people ahead.

What is parity, and why do pairing rules work?

Definition: Parity is the property of a number being even or odd. An even number can be arranged completely in pairs. An odd number leaves one object unpaired.

A pair contains two objects. An odd number can be viewed as one more than a collection of complete pairs, or one less than another collection of complete pairs. Pairing gives a reason for addition rules, rather than merely checking examples.

Property: Adding even numbers gives an even sum

A sum is the result of addition. Combining collections made entirely of pairs leaves all objects paired. Therefore, adding any number of even numbers gives an even number. The number of even numbers added does not change this conclusion.

Property: Two odd numbers have an even sum

Each odd collection contains pairs and one leftover object. When two odd collections are combined, their two leftover objects form another pair. Nothing remains unpaired, so odd + odd = even. Here, + means addition and = means “is equal to”.

Green and purple dots form paired columns. The separate odd collections each have an extra dot; in the combined collection, those extra dots make a pair.

Reference: NCERT Class 7, page 130, unnumbered diagram

Property: An even number plus an odd number is odd

The even collection contributes complete pairs. The odd collection contributes complete pairs and one leftover. Combining them leaves that object unpaired, giving even + odd = odd. These pairing arguments are proofs: explanations showing why the results hold.

For several odd numbers, pair the odd numbers with one another. An even number of odd numbers has an even sum. An odd number of odd numbers has an odd sum because one odd number remains after forming the pairs.

How can parity show that a puzzle is impossible?

Parity reasoning can rule out a proposed total before we calculate every possibility. First identify which quantities are odd and which are even. Then use the addition rules to find the required parity of the result.

Worked example 2. Kishor has five boxes and number cards that all show odd numbers. Exactly one card must go in each box. Can the five numbers add to 30?

Answer: No. Four of the five odd numbers can be grouped into two pairs of odd numbers. Each pair has an even sum. Adding the fifth odd number makes the total odd, whereas 30 is even.

Why can consecutive ages not total 112?

Consecutive numbers follow one another without a gap. Martin and Maria were born exactly one year apart and celebrate their birthday today. Their ages in completed years are therefore consecutive numbers, one even and the other odd.

Their sum must be odd, so it cannot be 112. Trying 51 and 52 gives 103, but the decisive argument is not this single trial. Every pair of consecutive counting numbers contains one odd number and one even number.

Worked example 3. Lakpa has an odd number of ₹1 coins, an odd number of ₹5 coins, and an even number of ₹10 coins. Could their total value be ₹205? The symbol ₹ denotes rupees.

Answer: The ₹1 coins have an odd total value, and an odd number of ₹5 coins also has an odd total value. The ₹10 coins contribute an even value. Odd plus odd plus even is even, so ₹205 is impossible.

These conclusions do not require the exact ages or coin counts. The given odd or even information already determines whether the proposed result can occur. An impossible parity is enough to reject a claim.

How do subtraction, multiplication and expressions affect parity?

The difference is the result of subtraction, written using −. Subtracting numbers of the same parity gives an even difference. Subtracting numbers of different parity gives an odd difference. This complements the pairing rules for addition.

What does the size of a grid tell us?

A grid is an arrangement of small squares in rows and columns. Rows run across; columns run down. In 3 × 4, the symbol × means multiplication: three rows of four squares contain 12 squares. This result is called a product; the numbers multiplied are its factors.

A 3 × 3 grid contains 9 squares. Both dimensions are odd, and the product is odd. More generally, a product of two odd numbers is odd. If at least one dimension is even, the squares can be paired and the product is even.

Worked example 4. Find the parity of the number of squares in grids of sizes 27 × 13, 42 × 78, and 135 × 654 without multiplying them out.

Answer: 27 and 13 are both odd, so 27 × 13 is odd. Both 42 and 78 are even, so their product is even. Since 654 is even, 135 × 654 is also even.

Can an expression change its parity?

A letter-number is a letter representing a number. An algebraic expression combines numbers, letter-numbers and operations. Let n represent an integer, meaning a whole number, its negative, or zero. In 3n + 4, writing 3n means 3 × n.

Thus, 3n + 4 can be odd or even. In contrast, 100p and 48w − 2 are even for integer values of the letter-numbers p and w. Multiplication by an even number produces an even value, and subtracting 2 preserves that parity.

How do we find an even or odd number at a given position?

To describe positions in a sequence, let n now be a positive counting number: 1, 2, 3, and so on. The phrase nth term means the term at position n. A formula gives that term directly from its position.

The positive even numbers are 2, 4, 6, 8, 10, 12, and so on. Each is twice its position. Therefore, nth even number = 2n. These are multiples of 2, meaning numbers obtained by multiplying 2 by counting numbers.

How are the two sequences related?

The positive odd numbers are 1, 3, 5, 7, 9, 11, and so on. Compare the sequences position by position. Each odd number is one less than the even number at the same position, so nth odd number = 2n − 1.

Worked example 5. Find the 100th positive even number and the 100th positive odd number.

Answer: Use n = 100. The even number is 2 × 100 = 200. Subtract 1 to find the odd number in the corresponding position: 200 − 1 = 199.

Does every even-valued expression list all even numbers?

No. Let k represent a positive counting number. The expression 6k + 2 gives 8, 14, 20, and so on when k takes the values 1, 2, 3, and so on. Its values are even, but many even numbers are missing.

Distinguish producing numbers of a particular parity from producing every number in that parity sequence. The formula 2n lists the positive even numbers in order. The formula 2n − 1 does the same for the positive odd numbers.

What can row and column sums reveal about a grid?

Consider a 3 × 3 grid filled with the numbers 1 to 9, using each exactly once. A row sum is the total of the three entries across a row. A column sum is the total of the three entries down a column.

Before filling such a grid, examine the proposed sums. A row or column contains three different numbers. The smallest possible sum is 1 + 2 + 3 = 6. The largest possible sum is 9 + 8 + 7 = 24.

Worked example 6. A grid must contain the numbers 1 to 9 exactly once. Its proposed sums include 5 for a row and 26 for a column. Can the grid be completed?

Answer: No. A sum of 5 is smaller than the minimum possible sum, 6. A sum of 26 exceeds the maximum possible sum, 24. Either condition is sufficient to make the proposed grid impossible.

Why must all three row sums total 45?

Adding the three row sums counts every entry once. Therefore, the result is 1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + 9 = 45. Adding the three column sums also counts every entry once and gives 45.

If all six row and column sums are added together, every entry is counted twice. Their combined total is therefore 90. These totals depend on using each of the numbers 1 to 9 exactly once, not on its position.

Use these observations to check consistency before trying placements. They give necessary conditions: requirements that every valid grid must satisfy. They can expose impossible proposals without a long search through different arrangements.

How is a magic square constructed using the numbers 1 to 9?

Definition: A magic square is a square grid whose rows, columns and two diagonals all have the same sum. This common total is its magic sum. A diagonal runs from one corner through the centre to the opposite corner.

Why must the magic sum be 15?

When the numbers 1 to 9 are used once each, the three row sums total 45. In a magic square these sums are equal. Thus, each is 45 ÷ 3 = 15, where ÷ means division. The columns and diagonals must also total 15.

Which number belongs in the centre?

The centre must contain 5. Testing alternatives helps reveal the restrictions. If 9 were at the centre, the line passing through 8 and 9 would already exceed 15 before adding its third positive number.

If 1 were at the centre, the line through 1 and 2 would need a third number outside the allowed range 1 to 9. Continuing this reasoning with other candidates leads to 5 as the required centre.

Why cannot 1 and 9 occupy corners?

A corner belongs to a row, a column and a diagonal. Putting 1 there would require three different pairs to accompany it to make 15. Only the pairs 5 and 9, or 6 and 8, work. Similar reasoning excludes 9 from corners.

Therefore, 1 and 9 occupy middle boundary positions, meaning the centres of the square's outer sides. They lie opposite one another through the centre 5. One completed arrangement is shown below.

Check each direction rather than just one row. For example, 8 + 1 + 6 = 15, 8 + 3 + 4 = 15, and 8 + 5 + 2 = 15. The other rows, columns and diagonal must meet the same requirement.

Ignoring rotations, which turn the square, and reflections, which produce mirror arrangements, there is exactly one such magic square using 1 to 9.

How can a magic square be changed or generalised?

Apply the same operation to every entry of a magic square. Adding 1 to each entry of a 3 × 3 magic square increases every row, column and diagonal sum by 3. Each contains three entries, so the totals remain equal.

Similarly, doubling every entry doubles every line sum. Subtracting the same number from every entry also preserves equal sums. Dividing every entry by the same non-zero number preserves equal sums as well.

Worked example 7. Start with the magic square whose rows are 8, 1, 6; 3, 5, 7; and 4, 9, 2. Add 1 to every entry to obtain a magic square using 2 to 10.

Answer: The new rows are 9, 2, 7; 4, 6, 8; and 5, 10, 3. Its magic sum is 15 + 3 = 18. Doubling the original entries instead gives a magic sum of 30.

How does the centre describe the whole square?

Let m represent the centre number. In one generalised arrangement using nine consecutive numbers, the other entries are expressed by adding to or subtracting from m. The following pattern retains the arrangement of the square used above.

In each row, column and diagonal, the added and subtracted amounts cancel. Three copies of m remain. Thus, magic sum = 3m for this generalised square. The letters describe relationships between entries, allowing many squares to share one pattern.

Worked example 8. Use the generalised pattern to form a magic square whose magic sum is 60.

Answer: Since 3m = 60, the centre is m = 20. The rows are 23, 16, 21; 18, 20, 22; and 19, 24, 17. Each row, column and diagonal totals 60.

Magic squares need not use consecutive numbers. For instance, multiplying every entry of the original square by the same number preserves its equal line sums, even when the resulting entries have gaps between them.

Where do magic squares appear in history and culture?

The Lo Shu Square, the first recorded magic square, dates back over 2000 years to ancient China. A legend describes a flood on the Lo River and a turtle sent by the gods, carrying a magical arrangement of the numbers 1 to 9.

Keep the account of the turtle as a legend. Magic squares were studied in India, Japan, Central Asia and Europe at different times. Indian mathematicians developed general construction methods for 3 × 3 and 4 × 4 squares, and also for larger squares.

What is the Chautīsā Yantra?

The Chautīsā Yantra is the first ever recorded 4 × 4 magic square. It appears in a 10th century inscription at the Pārśhvanath Jain temple in Khajuraho, India. Its name refers to 34, its magic sum.

Photograph: Chautīsā Yantra with its numerical grid (NCERT Class 7, page 137, unnumbered figure). A photograph of the inscribed square appears beside a printed four-row grid. The printed rows read 7, 12, 1, 14; 2, 13, 8, 11; 16, 3, 10, 5; and 9, 6, 15, 4.

Every row, column and diagonal in this square totals 34. Magic squares also appear in homes and shops in India, including the Navagraha Yantra. These examples connect numerical arrangements with cultural objects as well as mathematical investigation.

How does counting rhythms produce the Virahāṅka sequence?

In the poetry of many Indian languages, a syllable, a spoken unit of a word, is classified as short or long. A short syllable lasts one beat, a unit of time in a rhythm. A long syllable lasts two beats, exactly twice as long.

A rhythm here is an ordered arrangement of these short and long syllables. Counting rhythms is therefore equivalent to counting ways of writing a given number as a sum of 1s and 2s. The order matters: 1 + 2 and 2 + 1 describe different rhythms.

What happens for small numbers of beats?

Let n represent the number of beats. List the possibilities systematically, separating arrangements that start with a short syllable from those that start with a long one. The first four totals give the following complete lists.

Worked example 9. There are 5 rhythms with four beats and 3 rhythms with three beats. How many rhythms have five beats, if each syllable lasts one or two beats?

Answer: Put a one-beat syllable before each four-beat rhythm to get 5 possibilities. Put a two-beat syllable before each three-beat rhythm to get 3 more. Every five-beat rhythm starts in one of these ways, so there are 5 + 3 = 8.

Why does adding the previous two counts work?

The two groups have different first syllables, so they do not overlap. They also cover every possibility. For six beats, the corresponding count is 8 + 5 = 13. Continuing gives 1, 2, 3, 5, 8, 13, 21, 34, the Virahāṅka sequence.

The eight-beat count is 34. The same reasoning applies to Angaan's eight-step staircase when he can climb one or two steps at a time. Each ordered sequence of steps corresponds to a sequence of short and long syllables.

How do Virahāṅka-Fibonacci numbers connect patterns, history and nature?

In the Virahāṅka-Fibonacci sequence used here, the starting numbers are 1 and 2. Each later term is the sum of the two immediately preceding terms. For example, after 34 and 55 comes 34 + 55 = 89.

Worked example 10. Two consecutive terms are 987 and 1597. Find the next two and the previous two terms.

Answer: Add forwards: 987 + 1597 = 2584, followed by 1597 + 2584 = 4181. Subtract backwards: 1597 − 987 = 610, followed by 987 − 610 = 377. In sequence order, the previous terms are 377 and 610.

What parity pattern appears?

The parities begin odd, even, odd, odd, even, odd. The group odd, even, odd repeats. This follows from the addition rules: odd plus even gives odd, even plus odd gives odd, and odd plus odd gives even.

Thus, parity can be predicted without calculating large terms. The 20th term is even because its position lies second within a repeating group of three. Use the starting sequence 1, 2 when assigning these positions.

Who studied these numbers?

The Prakrit scholar Virahāṅka gave the rhythm-counting method around 700 CE, where CE means Common Era. He was the first known person to explicitly consider these numbers and state their formation rule. He expressed the method in a Prakrit poem.

His work drew on Piṅgala, who lived around 300 BCE, meaning Before Common Era. Gopala wrote about the numbers around 1135 CE and Hemachandra around 1150 CE. Fibonacci wrote about them in 1202 CE, about 500 years after Virahāṅka.

How do the numbers appear in nature?

These numbers connect poetry, drumming, visual arts, architecture and science. Perhaps their most stunning occurrences are in nature. The number of petals on a daisy is generally a Virahāṅka number; this is not a claim about every daisy.

Photographs: Daisies with 13, 21 and 34 petals. The three flowers illustrate the occurrence of Virahāṅka numbers in nature.

How do cryptarithms reveal digits hidden by letters?

Definition: A cryptarithm, also called an alphametic, is an arithmetic puzzle in which letters replace digits. A digit is one of the symbols 0 to 9. Each letter stands for a particular digit throughout the calculation.

Use place value, the value a digit has because of its position. In a two-digit number, the left digit represents tens and the right digit represents units. A three-digit number also has a hundreds position to the left.

How should adjacent letters be read?

In these puzzles, UT means a two-digit number with U in the tens place and T in the units place. It does not mean U multiplied by T. Each puzzle must be read according to its stated number of digits.

Worked example 11. Solve T + T + T = UT, where T is a one-digit number and UT is a two-digit result ending in the same digit T.

Answer: T = 5 gives 5 + 5 + 5 = 15, so U = 1. Algebraically, 3 × T = 10 × U + T, giving 2 × T = 10 × U. The two-digit result excludes T = 0.

How can the units column determine a repeated digit?

Consider K2 + K2 = HMM. K2 is a two-digit number with tens digit K and units digit 2. HMM is a three-digit number with hundreds digit H and the same digit M in both other positions.

Worked example 12. Solve K2 + K2 = HMM using the positions of the digits.

Answer: The units give 2 + 2 = 4, so M = 4. The result has three digits, and doubling a two-digit number gives a hundreds digit of 1, so H = 1. The result is 144, giving K = 7: 72 + 72 = 144.

A carry is an amount transferred to the next place when a column total reaches ten or more. Here, 7 tens plus 7 tens makes 14 tens: 1 hundred and 4 tens. That explains the hundreds digit and the repeated 4.

Finally, replace every letter with its proposed digit and check the complete calculation. A choice that works in the units column must also satisfy the tens and hundreds columns and the required number of digits.

Glossary

• Sequence — An ordered list of numbers whose positions distinguish its individual terms.
• Term — An individual number or expression occupying a particular position in a sequence.
• Parity — The property of an integer being either even or odd.
• Even number — A number of objects that can be arranged completely in pairs without leftovers.
• Odd number — A number of objects that leaves one object unpaired when arranged in pairs.
• Consecutive numbers — Numbers that follow one another in order without any intervening counting number.
• Letter-number — A letter used to represent a number in an algebraic expression.
• Row sum — The total obtained by adding all entries across one row of a grid.
• Column sum — The total obtained by adding all entries down one column of a grid.
• Magic square — A square grid whose rows, columns and two diagonals have equal sums.
• Magic sum — The common total of each row, column and diagonal of a magic square.
• Virahāṅka sequence — The sequence starting with 1 and 2, with each subsequent term adding the previous two.
• Cryptarithm — An arithmetic puzzle in which letters stand for particular digits that must be determined.

Common errors and misconceptions

• Misconception: A child saying 0 must be the tallest in the group. Correct: It means that no taller child is in front; a taller child may stand behind.
• Misconception: Any sum containing odd numbers is odd. Correct: An even number of odd numbers has an even sum because the leftover objects pair up.
• Misconception: An expression giving even values must list every positive even number. Correct: For positive counting values of k, 6k + 2 gives 8, 14, 20, and so on, missing many even numbers.
• Misconception: Equal row sums alone make a grid a magic square. Correct: Every column and both diagonals must also have that same sum.
• Misconception: Adding 1 to every entry raises a 3 × 3 magic sum by 1. Correct: Each line contains three entries, so its sum increases by 3.
• Misconception: The rhythms 1 + 2 and 2 + 1 are identical because both total 3. Correct: Their syllable orders differ, so they are different rhythms.
• Misconception: Every daisy must have a Virahāṅka number of petals. Correct: A daisy generally has a Virahāṅka number of petals, but this need not hold for every daisy.
• Misconception: UT in a cryptarithm means U multiplied by T. Correct: It denotes the two-digit number with tens digit U and units digit T.

Exam-style questions with model answers

Q1. In a line of 8 people of different heights, each person calls out the number of taller people in front. What is the largest possible call, and which placement achieves it? [2 marks]

1. The largest possible call is 7, since no person can have more than the other seven people in front.
2. Place the shortest person last. All seven people ahead are then taller, so that person calls out 7.

Q2. Kishor must put exactly one odd-number card in each of five boxes. Explain why the five numbers cannot total 30. [3 marks]

1. Group four of the five odd numbers into two pairs. Each pair has an even sum, since the two unpaired objects from its odd numbers form another pair.
2. Adding those two even sums still gives an even number. One odd number, the fifth card, remains to be added.
3. Even plus odd is odd. Thus the total must be odd, whereas 30 is even, making the required arrangement impossible.

Q3. Lakpa has an odd number of ₹1 coins, an odd number of ₹5 coins, and an even number of ₹10 coins. He reports a total of ₹205. Use parity to assess his claim. [4 marks]

1. An odd number of ₹1 coins has an odd total value, since multiplying an odd count by 1 leaves an odd number.
2. An odd number of ₹5 coins also has an odd total value, because odd multiplied by odd is odd.
3. The ₹10 coins contribute an even value. Combining the three groups gives odd plus odd plus even, which is even.
4. ₹205 is odd, so it cannot be the total under these conditions. Lakpa has made a mistake.

Q4. For the positive even sequence 2, 4, 6, 8, and so on, and the positive odd sequence 1, 3, 5, 7, and so on, find the 100th term of each. [2 marks]

1. The even term is twice its position, so the 100th positive even number is 2 × 100 = 200.
2. The corresponding odd term is one less, giving the 100th positive odd number as 200 − 1 = 199.

Q5. A 3 × 3 grid must use each number from 1 to 9 exactly once. Find its smallest and largest possible row sums, the total of all row sums, the total of all six row and column sums, and its magic sum if it is a magic square. Explain each result. [5 marks]

1. The smallest possible row sum is 1 + 2 + 3 = 6. These are the three smallest different entries allowed in the grid.
2. The largest possible row sum is 9 + 8 + 7 = 24. No other choice of three different allowed entries can give a larger sum.
3. All three row sums together total 45, because adding them counts each of the numbers 1 to 9 exactly once.
4. The three column sums also total 45. Adding all six row and column sums therefore gives 45 + 45 = 90, counting every entry twice.
5. If the grid is a magic square, its three row sums are equal. They must each be 45 ÷ 3 = 15, which is the magic sum.

Q6. A magic square has rows 8, 1, 6; 3, 5, 7; and 4, 9, 2, with magic sum 15. Add 1 to every entry. Write the new rows, find the new magic sum, and explain why the square remains magic. [3 marks]

1. The new rows are 9, 2, 7; 4, 6, 8; and 5, 10, 3. Each entry is exactly one greater than before.
2. Every row, column and diagonal contains three entries, each increased by 1. Therefore, each line total increases by 3, giving a new magic sum of 18.
3. All the required line sums undergo the same increase. They remain equal, so the resulting grid is still a magic square.

Q7. A short syllable lasts one beat and a long syllable two beats. Different orders count as different rhythms. Given 3 rhythms with three beats and 5 with four beats, find the counts for five, six, seven and eight beats, explaining the counting rule. [5 marks]

1. Every five-beat rhythm begins with either a short syllable followed by four beats, or a long syllable followed by three beats.
2. These groups do not overlap and cover all possibilities. Thus there are 5 + 3 = 8 rhythms with five beats.
3. The same split by the first syllable applies to six beats. Its count is the five-beat count plus the four-beat count: 8 + 5 = 13.
4. For seven beats, add the six-beat and five-beat counts. This gives 13 + 8 = 21 possible rhythms.
5. For eight beats, add the seven-beat and six-beat counts: 21 + 13 = 34. Hence there are 34 rhythms under the stated rules.

Q8. Solve K2 + K2 = HMM. K2 is a two-digit number ending in 2; HMM is a three-digit number whose tens and units digits are both M. Each letter stands for a particular digit. Explain using place value. [4 marks]

1. In the units column, 2 + 2 = 4 with no carry. Therefore, the repeated digit M must be 4.
2. Doubling a two-digit number can produce a three-digit result only with hundreds digit 1. Hence H = 1, making the result 144.
3. Half of 144 is 72. Since K2 is 72, its tens digit is K = 7.
4. Substitution checks the complete answer: 72 + 72 = 144. Thus K = 7, H = 1 and M = 4 satisfy every column.

Key takeaways

• The height rule counts taller people in front, so a call of zero need not identify the tallest person.
• Pairing explains parity: two odd numbers have an even sum, while an even number plus an odd number is odd.
• A product is even if at least one factor is even; multiplying two odd numbers gives an odd product.
• At positive counting position n, the even number is 2n and the odd number is 2n − 1.
• Using 1 to 9 once in a magic square requires a magic sum of 15 and a centre of 5.
• Adding the same amount to every entry preserves equal line sums; doubling every entry doubles the magic sum.
• The Virahāṅka sequence starts 1, 2 and continues by adding the previous two terms, counting rhythms of short and long syllables.
• Cryptarithms use fixed digits for letters, with place value and column calculations helping reveal the hidden numbers.

Test yourself

Why does the first person in the taller-people-in-front activity say 0?

Nobody stands in front of the first person, so there are no taller people ahead to count.

What is the parity of the sum of eight odd numbers?

It is even. Group the eight odd numbers into four pairs; each pair has an even sum.

Is the number of squares in a 135 × 654 grid odd or even?

It is even because 654 is even, and a product with an even factor is even.

Can a row sum be 26 in a 3 × 3 grid using each number from 1 to 9 once?

No. The largest possible sum of three different allowed numbers is 9 + 8 + 7 = 24.

What happens to a 3 × 3 magic sum of 15 when every entry increases by 1?

It becomes 18 because each row, column and diagonal contains three entries, each increased by one.

The Virahāṅka sequence starts 1, 2. What parity pattern repeats?

The repeating group is odd, even, odd, as follows from adding each pair of preceding terms.

A bulb is initially on. Dorjee toggles its switch 77 times, changing between on and off each time. What is its final state?

It is off. Every pair of toggles restores the starting state; 77 is odd, leaving one extra toggle.

In the cryptarithm T + T + T = UT, where UT is a two-digit number, what are T and U?

T is 5 and U is 1, because 5 + 5 + 5 = 15.