Correct option 3:5

Analyzing the Number Arrangement Machine

The given problem describes a machine that transforms a line of numbers through a sequence of steps, following specific rules. We need to deduce these rules by observing the provided input and its rearrangement in four steps.

Rule for Step I

Let's look at the transformation from Input to Step I:

• Input: 67543 > Step I: 42412
• Input: 49652 > Step I: 36510
• Input: 86235 > Step I: 48115
• Input: 27549 > Step I: 14436
• Input: 59246 > Step I: 45124
• Input: 38673 > Step I: 24521

Observing the numbers, let's consider a 5-digit input number ABCDE. The output in Step I seems to be formed by concatenating three parts:

1. The product of the first two digits (A*B).
2. The middle digit minus one (C-1).
3. The product of the last two digits (D*E).

Let's test this rule:

• 67543: (6*7) = 42, (5-1) = 4, (4*3) = 12. Concatenate: 42412. (Matches)
• 49652: (4*9) = 36, (6-1) = 5, (5*2) = 10. Concatenate: 36510. (Matches)
• 86235: (8*6) = 48, (2-1) = 1, (3*5) = 15. Concatenate: 48115. (Matches)
• 27549: (2*7) = 14, (5-1) = 4, (4*9) = 36. Concatenate: 14436. (Matches)
• 59246: (5*9) = 45, (2-1) = 1, (4*6) = 24. Concatenate: 45124. (Matches)
• 38673: (3*8) = 24, (6-1) = 5, (7*3) = 21. Concatenate: 24521. (Matches)

The rule for Step I is confirmed: For a 5-digit number ABCDE, the output is the concatenation of (A*B), (C-1), and (D*E).

Rule for Step II

Now let's analyze the transformation from Step I to Step II:

• Step I: 42412 > Step II: 14422
• Step I: 36510 > Step II: 53160
• Step I: 48115 > Step II: 51184
• Step I: 14436 > Step II: 31644
• Step I: 45124 > Step II: 51442
• Step I: 24521 > Step II: 51422

Let's examine the digits in Step I numbers and their corresponding Step II numbers. For 42412 (Step I) to 14422 (Step II), the digits are {4, 2, 4, 1, 2} transformed into {1, 4, 4, 2, 2}. Let's separate the odd and even digits from Step I numbers:

• 42412: Odd={1}, Even={4, 2, 4, 2}. If we arrange odd digits descending (1) followed by even digits descending (4, 4, 2, 2), we get 14422. (Matches)
• 36510: Odd={3, 5, 1}, Even={6, 0}. Descending Odd: 5, 3, 1. Descending Even: 6, 0. Concatenate: 53160. (Matches)
• 48115: Odd={1, 1, 5}, Even={4, 8}. Descending Odd: 5, 1, 1. Descending Even: 8, 4. Concatenate: 51184. (Matches)
• 14436: Odd={1, 3}, Even={4, 4, 6}. Descending Odd: 3, 1. Descending Even: 6, 4, 4. Concatenate: 31644. (Matches)
• 45124: Odd={5, 1}, Even={4, 2, 4}. Descending Odd: 5, 1. Descending Even: 4, 4, 2. Concatenate: 51442. (Matches)
• 24521: Odd={5, 1}, Even={2, 4, 2}. Descending Odd: 5, 1. Descending Even: 4, 2, 2. Concatenate: 51422. (Matches)

The rule for Step II is confirmed: For each number in Step I, arrange its odd digits in descending order, followed by its even digits in descending order, and concatenate them to form the Step II number.

Applying the Rules to the New Input

Now, let's apply these rules to the given input:

Input : 45483   45192   35654   79583   43235   55672

Step I Calculation

Using the rule (A*B)(C-1)(D*E):

1. 45483: (4*5)(4-1)(8*3) = 20   3   24 > 20324
2. 45192: (4*5)(1-1)(9*2) = 20   0   18 > 20018
3. 35654: (3*5)(6-1)(5*4) = 15   5   20 > 15520
4. 79583: (7*9)(5-1)(8*3) = 63   4   24 > 63424
5. 43235: (4*3)(2-1)(3*5) = 12   1   15 > 12115
6. 55672: (5*5)(6-1)(7*2) = 25   5   14 > 25514

Step I: 20324   20018   15520   63424   12115   25514

Step II Calculation

Using the rule: Odd digits descending, followed by Even digits descending.

1. 20324: Odd={3}, Even={2,0,2,4}. Descending Odd: 3. Descending Even: 4,2,2,0. > 34220
2. 20018: Odd={1}, Even={2,0,0,8}. Descending Odd: 1. Descending Even: 8,2,0,0. > 18200
3. 15520: Odd={1,5,5}, Even={2,0}. Descending Odd: 5,5,1. Descending Even: 2,0. > 55120
4. 63424: Odd={3}, Even={6,4,2,4}. Descending Odd: 3. Descending Even: 6,4,4,2. > 36442
5. 12115: Odd={1,2,1,1,5}. Correction: Odd={1,1,1,5}, Even={2}. Descending Odd: 5,1,1,1. Descending Even: 2. > 51112
6. 25514: Odd={2,5,5,1,4}. Correction: Odd={5,5,1}, Even={2,4}. Descending Odd: 5,5,1. Descending Even: 4,2. > 55142

Step II: 34220   18200   55120   36442   51112   55142

Finding the Required Values in Step II

The question asks for the sum of the largest digit of the first number and the smallest digit of the last number in Step II.

• First number in Step II: 34220
• Last number in Step II: 55142

Largest digit of the first number (34220): The digits are 3, 4, 2, 2, 0. The largest digit is 4.

Smallest digit of the last number (55142): The digits are 5, 5, 1, 4, 2. The smallest digit is 1.

Calculating the Sum

Sum = (Largest digit of first number in Step II) + (Smallest digit of last number in Step II)

Sum = 4 + 1 = 5

The sum is 5.