A typical football is made by stitching together 12 pentagons and 20 hexagons. How many vertices (junctions) are there in such a football?

How many matching pairs of shoes (both left and right shoe) are present in the image below? [Figure: see page 5 of source PDF]

A wooden block of dimension 10cm x 20cm x 30cm is cut into equal sized planks. The cut planks are stacked one above the other to achieve a total height of 100cm exactly. If the minimum number of planks are cut to achieve this height, then what is the volume of each plank in cm³?

A perspective view of a solid object is shown on the left. The object is cut simultaneously along THREE perpendicular planes as shown on the right. How many surfaces will the resulting pieces have in total (i.e. sum of the surfaces of all pieces)? [Figure: see page 4 of source PD…

Shown below are two stacked paper cups and a box with their dimensions. If stacking is allowed, then what is the maximum number of cups that can be stored in the box without deforming the cups? [Figure: see page 6 of source PDF]

A perspective view of a solid is shown below. The solid is symmetrical, and hidden surfaces such as the base are flat. What is the total number of surfaces in the solid? [Figure: see page 4 of source PDF]

Section P shows three views of a regular dice. TEN of such regular dice are stacked on an opaque table as shown in Section Q (two views of the same arrangement). What is the maximum sum that can be achieved adding the numbers on the visible surfaces from all angles? [Figure: see…

Given below are ten pieces of a puzzle. When arranged correctly they form a four-digit number. What is the number formed after the correct arrangement? [Figure: see page 4 of source PDF]

What are the minimum number of tile pieces (shown on the right) that are required to create a path from tile A to tile B, such that ALL the following conditions are met: (1) All tiles are to be used at least once. (2) Tiles cannot overlap. (3) The path on a tile must be connected…