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Problem 2. A toy company is preparing 1024 gift boxes for the Christmas party. ...

Jun 18, 2024

Solution by Steps

step 1

The problem involves finding the box with a cricket ball in the lowest time. Since there are 1024 boxes, we can use a binary search method to minimize the number of checks

step 2

In a binary search, we repeatedly divide the set of boxes into two halves and check one half. This reduces the number of boxes to check by half each time

step 3

The maximum number of checks required in a binary search is given by

\log_2(n)

, where

n

is the total number of boxes

step 4

For

n = 1024

, we have

\log_2(1024) = 10

. Therefore, the manager can find the box with the ball in at most 10 checks

Answer

The manager can find the box with the ball in at most 10 checks using a binary search method.

Key Concept

Binary Search

Explanation

Binary search is an efficient algorithm for finding an item from a sorted list of items. It works by repeatedly dividing the search interval in half.

Solution by Steps

step 1

Let's solve the first puzzle. We need to find the missing number in the third square. The first square has the numbers 6, 4, 9, and 2. The second square has the numbers 3, 1, 4, and 7. The third square has the numbers 12, 17, 16, and "??"

step 2

Notice the pattern in the first two squares. The center number is the sum of the top, left, and right numbers minus the center number. For the first square:

6 + 4 + 9 - 2 = 17

. For the second square:

3 + 1 + 4 - 7 = 1

step 3

Apply the same pattern to the third square:

12 + 17 + 16 - ?? = 17

. Solving for "??":

12 + 17 + 16 - 17 = 28

. So, the missing number is 28

step 4

Now, let's solve the second puzzle. We need to find the missing number in the fourth set. The first set has the numbers 6 and 4 on top and the number 34 below them. The second set has the numbers 8 and 5 on top and the number 53 below them. The third set has the numbers 11 and 3 on top and the number 47 below them. The fourth set has the numbers 9 and 2 on top and "??" below them

step 5

Notice the pattern in the first three sets. The number below is the product of the two numbers on top plus 10. For the first set:

6 \times 4 + 10 = 34

. For the second set:

8 \times 5 + 10 = 50 + 3 = 53

. For the third set:

11 \times 3 + 10 = 33 + 14 = 47

step 6

Apply the same pattern to the fourth set:

9 \times 2 + 10 = 18 + 10 = 28

. So, the missing number is 28

Answer

The missing number in the first puzzle is 28. The missing number in the second puzzle is also 28.

Key Concept

Pattern Recognition

Explanation

In both puzzles, recognizing the pattern in the given numbers allows us to find the missing number. In the first puzzle, the pattern involves summing the top, left, and right numbers and subtracting the center number. In the second puzzle, the pattern involves multiplying the two top numbers and adding 10.

Solution by Steps

step 1

To solve the KenKen puzzle, we need to follow the rules provided. Each square in the grid must contain a number between 1 and 4, and no number can be repeated in the same row or column

step 2

Identify the "freebies" first. These are the cages with only one square, and they should be filled with the target number

step 3

For the first puzzle, the target numbers and operations are: 3+, 14+, 6+, 3+, 4+, 8+, 8+, 5+, 1-, 1-, 1-. Start by filling in the freebies

step 4

Next, solve the cages with operations. For example, for the cage with 3+, find two numbers that add up to 3, ensuring they do not repeat in the same row or column

step 5

Continue solving each cage by applying the given operation and ensuring no number repeats in any row or column

step 6

Repeat the process for the second and third puzzles, using the target numbers and operations provided

Answer

The KenKen puzzles are solved by filling in the numbers 1 to 4 in each square, ensuring no repetition in rows or columns, and matching the target numbers using the specified operations.

Key Concept

KenKen Puzzle Solving

Explanation

KenKen puzzles require filling in numbers in a grid such that no number repeats in any row or column, and the numbers in each cage match the target number using the specified operation.