4

Part of 2000 May Olympiad

Problems(2)

cubes of 1x1x1 in cube 3x3x3 related

Source: VI May Olympiad (Olimpiada de Mayo) 2000 L2 P4

There is a cube of

3 \times 3 \times 3

formed by the union of

27

1 \times 1 \times 1

. Some cubes are removed in such a way that those that remain continue to form a solid made up of cubes that are united by at least one facing the rest of the solid. When a cube is removed, those that remain do so in the same place they were. What is the maximum number of cubes that can be removed so that the area of the resulting solid is equal to the area of the original cube?

combinatoricscombinatorial geometry

min no of pieces, equilateral from equilaterals

Source: VI May Olympiad (Olimpiada de Mayo) 2000 L1 P4

There are pieces in the shape of an equilateral triangle with sides

1, 2, 3, 4, 5

6

50

pieces of each size). You want to build an equilateral triangle of side

7

using some of these pieces, without gaps or overlaps. What is the least number of pieces needed?

geometryEquilateralequilateralscombinatorial geometry