Solve this cryptarithm: UT + TA = TAT

Since each of the 50 sheets is printed on both sides, there are $50 \times 2 = 100$ pages.

Let the first page number be $n$. The pages are $n, n+1, \dots, n+99$.
The sum of an arithmetic series is given by $S = \frac{\text{number of terms}}{2} \times (\text{first term} + \text{last term})$.

We set the sum equal to 6000:

Divide both sides by 50:

Subtract 99 from both sides:

Divide by 2:

Since page numbers must be whole integers, $n$ cannot be 10.5. Therefore, the sum of the page numbers of the loose sheets cannot be 6000.

3. Here is a 2 x 3 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.

The grid has 2 rows and 3 columns.
Row 1 sum parity: Even (e)
Row 2 sum parity: Even (e)
Column 1 sum parity: Even (e)
Column 2 sum parity: Even (e)
Column 3 sum parity: Odd (o)

Total sum parity (from rows) = (Row 1 sum parity) + (Row 2 sum parity) = Even + Even = Even.

Total sum parity (from columns) = (Col 1 sum parity) + (Col 2 sum parity) + (Col 3 sum parity) = Even + Even + Odd = Odd.

The total sum of all elements in the grid must have a consistent parity, regardless of whether it's calculated by summing rows or summing columns.
However, we found that the total sum parity from rows is Even, and the total sum parity from columns is Odd.
Since Even $\neq$ Odd, there is a contradiction.

It is impossible to fill the 6 boxes with 3 odd numbers and 3 even numbers to satisfy the given row and column sum parities because the sum of the row parities does not match the sum of the column parities.

4. Make a 3 x 3 magic square with 0 as the magic sum. All numbers can not be zero. Use negative numbers, as needed.

In a 3x3 magic square, the sum of numbers in each row, column, and main diagonal is the same (the magic sum). For a 3x3 square, the center element is equal to the magic sum divided by 3.
Given magic sum = 0.
So, the center element $e = \frac{0}{3} = 0$.

Let the square be:

Since the sum of each row, column, and diagonal is 0:

• $d + 0 + f = 0 \implies f = -d$
• $b + 0 + h = 0 \implies h = -b$
• $a + 0 + i = 0 \implies i = -a$
• $c + 0 + g = 0 \implies g = -c$

We need $a+b+c = 0$ (Row 1 sum) and $a+d-c = 0$ (Column 1 sum).
We need to choose values such that not all numbers are zero.
Let's choose $a=1$.
From $a+d-c=0$, we have $1+d-c=0 \implies c = d+1$.
From $a+b+c=0$, we have $1+b+(d+1)=0 \implies b+d+2=0 \implies b = -d-2$.

Let's pick a value for $d$. Let $d=2$.
Then $c = 2+1 = 3$.
And $b = -2-2 = -4$.

$a=1$
$b=-4$
$c=3$
$d=2$
$e=0$
$f=-d = -2$
$g=-c = -3$
$h=-b = 4$
$i=-a = -1$

5. Fill in the following blanks with 'odd' or 'even':

a) Sum of an odd number of even numbers is
An even number is always of the form $2k$. The sum of any number of even numbers is always even.
Example: $2+4+6 = 12$ (Even).

b) Sum of an even number of odd numbers is
An odd number is always of the form $2k+1$. The sum of two odd numbers is even ($O+O=E$). If there is an even number of odd numbers, they can be paired up, and each pair sums to an even number. The sum of even numbers is even.
Example: $1+3+5+7 = 16$ (Even).

c) Sum of an even number of even numbers is
The sum of any number of even numbers is always even.
Example: $2+4 = 6$ (Even).

d) Sum of an odd number of odd numbers is
The sum of two odd numbers is even ($O+O=E$). If there is an odd number of odd numbers, all but one can be paired up, resulting in a sum of even numbers plus one remaining odd number. An even number plus an odd number is odd ($E+O=O$).
Example: $1+3+5 = 9$ (Odd).

6. What is the parity of the sum of the numbers from 1 to 100?

The sum $S$ of the first $n$ natural numbers is given by $S = \frac{n(n+1)}{2}$.
For $n=100$:

The number 5050 is divisible by 2, so it is an even number.
Alternatively, an even number (50) multiplied by an odd number (101) always results in an even number.

7. Two consecutive numbers in the Virahānka sequence are 987 and 1597. What are the next 2 numbers in the sequence? What are the previous 2 numbers in the sequence?

The Virahānka sequence is another name for the Fibonacci sequence, where each number is the sum of the two preceding ones ($F_n = F_{n-1} + F_{n-2}$).

Given $F_k = 987$ and $F_{k+1} = 1597$.
The next number, $F_{k+2}$, is the sum of the two given numbers:

The number after that, $F_{k+3}$, is the sum of $F_{k+2}$ and $F_{k+1}$:

To find the previous number, $F_{k-1}$, we subtract $F_k$ from $F_{k+1}$:

To find the number before that, $F_{k-2}$, we subtract $F_{k-1}$ from $F_k$:

8. Angaan 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?

This is a classic problem that can be solved using the Fibonacci sequence. Let $W(n)$ be the number of ways to climb $n$ steps.

• To reach step $n$, Angaan must have come from either step $n-1$ (by taking 1 step) or step $n-2$ (by taking 2 steps).
• So, the number of ways to reach step $n$ is $W(n) = W(n-1) + W(n-2)$.

• For 1 step: $W(1) = 1$ (path: 1)
• For 2 steps: $W(2) = 2$ (paths: 1,1; 2)

• $W(3) = W(2) + W(1) = 2 + 1 = 3$
• $W(4) = W(3) + W(2) = 3 + 2 = 5$
• $W(5) = W(4) + W(3) = 5 + 3 = 8$
• $W(6) = W(5) + W(4) = 8 + 5 = 13$
• $W(7) = W(6) + W(5) = 13 + 8 = 21$
• $W(8) = W(7) + W(6) = 21 + 13 = 34$

9. What is the parity of the 20th term of the Virahānka sequence?

The Virahānka (Fibonacci) sequence starts with $F_1=1, F_2=1$.
Let's list the terms and their parities:
$F_1 = 1$ (Odd)
$F_2 = 1$ (Odd)
$F_3 = F_1 + F_2 = 1+1 = 2$ (Even)
$F_4 = F_2 + F_3 = 1+2 = 3$ (Odd)
$F_5 = F_3 + F_4 = 2+3 = 5$ (Odd)
$F_6 = F_4 + F_5 = 3+5 = 8$ (Even)
$F_7 = F_5 + F_6 = 5+8 = 13$ (Odd)
$F_8 = F_6 + F_7 = 8+13 = 21$ (Odd)
$F_9 = F_7 + F_8 = 13+21 = 34$ (Even)

The pattern of parities is: Odd, Odd, Even, Odd, Odd, Even, ...
This pattern repeats every 3 terms (O, O, E).

To find the parity of the 20th term, we can find the remainder when 20 is divided by 3.
$20 \div 3 = 6$ with a remainder of 2.
This means the 20th term will have the same parity as the 2nd term in the repeating pattern (O, O, E).
The 2nd term in the pattern is Odd.

10. Identify the statements that are true.

a) The expression $4m - 1$ always gives odd numbers.

$4m$ is always an even number (since 4 is even, $4 \times \text{any integer}$ is even).
Subtracting 1 from an even number ($E - 1$) always results in an odd number.
Example: If $m=1$, $4(1)-1 = 3$ (Odd). If $m=2$, $4(2)-1 = 7$ (Odd).
This statement is True.

b) All even numbers can be expressed as $6j - 4$.

$6j - 4$ is always an even number (since $6j$ is even and 4 is even, $E-E=E$).

Let's test some even numbers.
If $j=1$, $6(1)-4 = 2$.
If $j=2$, $6(2)-4 = 8$.
If $j=0$, $6(0)-4 = -4$.
If $j=-1$, $6(-1)-4 = -10$.
This expression generates even numbers, but not all of them. For example, 4 cannot be expressed as $6j-4$ for any integer $j$ ($6j-4=4 \implies 6j=8 \implies j=8/6$, not an integer).
This statement is False.

c) Both expressions $2p + 1$ and $2q - 1$ describe all odd numbers.

$2p+1$ describes all odd numbers (where $p$ is any integer).
$2q-1$ also describes all odd numbers (where $q$ is any integer).
Example: For $p=0, 2(0)+1=1$. For $p=1, 2(1)+1=3$. For $p=-1, 2(-1)+1=-1$.
For $q=1, 2(1)-1=1$. For $q=2, 2(2)-1=3$. For $q=0, 2(0)-1=-1$.
Both expressions correctly generate all odd numbers.
This statement is True.

d) The expression $2f + 3$ gives both even and odd numbers.

$2f$ is always an even number (since 2 is even, $2 \times \text{any integer}$ is even).
Adding 3 (an odd number) to an even number ($E+O$) always results in an odd number.
Example: If $f=1$, $2(1)+3 =