Section A – Cryptarithms (Alphametics)
Solve each. Every letter represents a single digit. Same letters = same digit, different letters = different digits.
1. Find U and T.
UT+TA=TAT
2. In the alphametic below, each letter stands for one digit:
K2+K2=HMM
What digits do H and M represent?
3. Find the digits represented by the letters:
YY+Z=ZOO
4. Solve the cryptarithm:
B5+3D=ED5
5. Find P, R in the alphametic:
KP+KP=PRR
6. Find C and F:
C1+C=1FF
Section B – Parity & Logical Reasoning
7. A light bulb is ON. Dorjee toggles the switch 59 times. Will the bulb be ON or OFF at the end? Explain why.
8. 50 loose sheets (each sheet has 2 pages printed front and back) fell out of a book. Could the sum of the page numbers of these 50 sheets be 6000? Explain logically.
9. Fill the 2×3 grid using 3 even and 3 odd numbers so that the row & column sums match the given parity:
[ ] [ ] [ ] Row parity: eColumn parities: e o e
Fill with any integers that fit the parity rules.
10. Make a 3×3 magic square with magic sum 0. You may use negatives.
(Condition: not all numbers can be zero.)
11. Fill in the blanks with “odd” or “even”:
a. The sum of an odd number of even numbers is ________
b. The sum of an even number of odd numbers is ________
c. The sum of an even number of even numbers is ________
d. The sum of an odd number of odd numbers is ________
Section C – Patterns & Sequences
12. What is the parity (odd/even) of the sum of the numbers from 1 to 100?
13. 987 and 1597 are consecutive numbers in the Virahāṅka (Fibonacci) sequence.
Find:
a. The next two numbers
b. The previous two numbers
14. What is the parity (odd/even) of the 20th term of the Virahāṅka sequence?
Section D – Combinatorics
15. Angaan climbs an 8-step staircase. He can take steps of 1 or 2 at a time. In how many different ways can he reach the top?
Section E – Statements: True or False
16. Identify all true statements:
(a) 4m − 1 always gives odd numbers.
(b) All even numbers can be written as 6j − 4.
(c) Both expressions 2p + 1 and 2q − 1 describe all odd numbers.
(d) 2f + 3 can give both even and odd numbers.
Section F – More Number Puzzles
17. A number ends in 8. When doubled, the new number ends in 6. What could the last digit of the tens place have been?
18. Find the missing digit X:
The number 4X3 is divisible by 3 and 9.
19. Replace letters with digits to satisfy:
AB+BA=C00
What digits can A, B, C be?
20. You have digits 1, 2, 3, 4. Form a 2-digit number and a 3-digit number using all digits exactly once. Their sum should be the greatest possible. What is the sum?
21. A certain number leaves remainder 3 when divided by 5, 7, and 9. What is the smallest such number?
22. Fill the blanks:
The sum of three numbers is _____.
One is even, two are odd. Is the sum odd or even? Explain why.
23. What is the largest 3-digit number whose digits sum to 15 and the number is divisible by 3?
24. Fill the cryptarithm:
PQ+8=QP
Find P and Q.
25. A pattern goes: 3, 6, 10, 15, 21, …
Find the 15th term.
26. Find the digit X:
7X+X7=154
27. If AB × 4 = BA, find all possible 2-digit numbers AB.
28. Write all 3-digit palindromes that are divisible by 11.
29. A number is divisible by 9 and its digits add to 27. What is the smallest possible 4-digit number that fits?
30. How many 2-digit numbers have an even digit sum?
(Hint: think parity pairings of tens & units digits.)
