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One-line Exam Shortcut:

Look at the movement of digits:
61 → 52 → 63 → 94

First digit changes: −1, +1, +3
Second digit changes: +1, +1, +1

Continue the pattern:
First digit: 9 − 5 = 4
Second digit: 4 + 2 = 6

Next number = 46

Correct Answer: 46

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• 96 ÷ 2 = 48
• 48 ÷ 2 = 24

Below Half:

• 64 ÷ 2 = 32
• 32 ÷ 2 = 16

Thus, each step divides the number by 2 successively for both the upper and lower halves.

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Digit Puzzle: Final Digit of the Middle Number

Given Numbers: 589, 362, 554, 371, 442

Step 1: Subtract 3 from the middle digit of each number

• 589 → 589 → 559 → 559
• 362 → 362 → 332 → 332
• 554 → 554 → 524 → 524
• 371 → 371 → 341 → 341
• 442 → 442 → 412 → 412

Step 2: Reverse the digits of each new number

• 559 → 955
• 332 → 233
• 524 → 425
• 341 → 143
• 412 → 214

Step 3: Arrange in descending order

955, 425, 233, 214, 143

Step 4: Pick the middle number

Middle number = 233

Step 5: Find the last digit

Last digit of 233 = 3

✅ Final Answer: 3

Solution

Series: 3, 10, 24, 45, 73, __

Differences:

10−3 = 7
24−10 = 14
45−24 = 21
73−45 = 28

The differences are: 7, 14, 21, 28 → an arithmetic sequence with common difference +7. So the next difference is 35.

Next term: 73 + 35 = 108

Answer: 108

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Given
$A_1={3}$,
$A_2={5,7,9}$,
$A_3={11,13,15,17,19}$,
$A_4={21,23,25,27,29,31,33}$, …

Observation:

• $A_k$ contains $2k-1$ numbers
• All elements are consecutive odd numbers
• So the average of $A_k$ is simply its middle term

First term of $A_k$:
$3,5,11,21,\dots$

This follows
$a_k = 3 + 2(k-1)^2$

Middle term of $A_k$:
$\text{Average} = a_k + (2k-2)$

So,
$\text{Average} = 3 + 2(k-1)^2 + 2(k-1)$
$\text{Average} = 3 + 2k(k-1)$

For $k=20$:
$\text{Average} = 3 + 2\times20\times19$
$\text{Average} = 3 + 760$
$\boxed{763}$

Solution

Given rule: Add numbers in the base where 137 + 276 = 435 holds.

Find the base \(b\):

\(137_b = 1b^2 + 3b + 7\)
\(276_b = 2b^2 + 7b + 6\)
Sum \(= 3b^2 + 10b + 13\)

\(435_b = 4b^2 + 3b + 5\)

Equate: \(3b^2 + 10b + 13 = 4b^2 + 3b + 5 \;\Rightarrow\; b^2 - 7b - 8 = 0\)

So \(b = 8\) (since base must be > 7). Thus the operations are in base 8 (octal).

Now add in base 8: \(731_8 + 672_8\)

      731₈
    + 672₈
    -------
      (units) 1 + 2 = 3 → write 3
      (eights) 3 + 7 = 10₁₀ = 12₈ → write 2, carry 1
      (sixty-fours) 7 + 6 + 1 = 14₁₀ = 16₈ → write 6, carry 1
      bring down carry → 1
    -------
      1623₈
  

Answer: \(731 + 672 = \mathbf{1623_8}\) (in base 8).

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Series: 42, 40, 38, 35, 33, 31, 28, …

Pattern (differences): −2, −2, −3, −2, −2, −3, … (repeats)

After 28: 28 − 2 = 26, then 26 − 2 = 24.

Next pair: 26, 24

Solution

Series: 122, 144, 166, 188, ?

Differences:

144 − 122 = 22
166 − 144 = 22
188 − 166 = 22

The pattern is a constant increase of +22. So the next term is:

188 + 22 = 210

Answer: 210

Solution

Series: 6, 11, 21, 36, 56, ?

Differences

11 − 6 = 5
21 − 11 = 10
36 − 21 = 15
56 − 36 = 20

The differences are +5, +10, +15, +20 (increase by 5 each time). Next difference = +25.

Next term

56 + 25 = 81

Answer: 81

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Reason: The sequence subtracts consecutive odd numbers decreasing by 2 each step:

99 → 90 : −9
90 → 83 : −7
83 → 78 : −5
78 → 75 : −3 (next odd)
75 → 74 : −1 (next odd)

So after 78, the terms are: 75, 74.

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