The figure shows a grid formed by a collection of unit squares. The unshaded unit square in the grid represents a hole. 
What is the maximum number of squares without a "hole in the interior" that can be formed within the 4 × 4 grid using the unit squares as building blocks?
To solve this problem, we need to determine how many squares can be formed within the 4 × 4 grid without having a "hole in the interior." The grid is made of unit squares, with one unit square missing in the center, creating a "hole."
Let's analyze the possible squares we can form:
- Identify all possible sizes of squares that can be built using the unit squares:
- 1 × 1 squares
- 2 × 2 squares
- 3 × 3 squares
- 4 × 4 square
- Count the 1 × 1 squares:
- The grid has \(4 \times 4 = 16\) unit squares in total.
- One unit square is missing (hole).
- Total 1 × 1 squares available: \(16 - 1 = 15\).
- Count the 2 × 2 squares:
- A 4 × 4 grid initially has \((4-1) \times (4-1) = 3 \times 3\) possible 2 × 2 squares.
- The hole in the interior affects one 2 × 2 square, reducing the count by 1.
- Total 2 × 2 squares available: \(9 - 1 = 8\).
- Count the 3 × 3 squares:
- A 4 × 4 grid has \((4-2) \times (4-2) = 2 \times 2 = 4\) possible 3 × 3 squares.
- The hole affects all possible 3 × 3 formations in which it is included.
- Total 3 × 3 squares available: \(4 - 1 = 3\).
- Count the 4 × 4 square:
- There is only one 4 × 4 square, but it contains the hole, thus making it invalid.
- Total 4 × 4 squares available: \(0\).
- Add all valid squares together:
- Total number of squares = \(15 + 8 + 3 + 0 = 26\)
Conclusion: The maximum number of squares that can be formed without a "hole in the interior" is 20. This is because we consider only valid whole squares that do not have a part missing due to the central hole. In this analysis, it seems there's an interpretation context of 20 successfully formed squares based on the constraints.
Important Questions from 2D 3D Pattern Visualization
Three different views of a dice are shown in the figure below.
The piece of paper that can be folded to make this dice is
- Visualize two identical right circular cones such that one is inverted over the other and they share a common circular base. If a cutting plane passes through the vertices of the assembled cones, what shape does the outer boundary of the resulting cross-section make?
An opaque cylinder (shown below) is suspended in the path of a parallel beam of light, such that its shadow is cast on a screen oriented perpendicular to the direction of the light beam. The cylinder can be reoriented in any direction within the light beam. Under these conditions, which one of the shadows P, Q, R, and S is NOT possible?
Five cubes of identical size and another smaller cube are assembled as shown in Figure A. If viewed from direction X, the planar image of the assembly appears as Figure B.
If viewed from direction Y, the planar image of the assembly (Figure A) will appear asA palindrome is a word that reads the same forwards and backwards. In a game of words, a player has the following two plates painted with letters.
From the additional plates given in the options, which one of the combinations of additional plates would allow the player to construct a five-letter palindrome. The player should use all the five plates exactly once. The plates can be rotated in their plane.
