Slice the square into 4 identical sections, so that in each section there is 1 caterpillar with its leaf. One caterpillar will not have a leaf, she is taking a diet.

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Caterpillars - solution

Slice the picture into 2 sections (Edit: jagged cut) from which you could make a rectangle 6x4 squares.

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6x4 Rectangle - solution

Slice the picture into 4 identical sections.

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4 Identical Sections - solution

Divide a paper into 8 sections and write numbers on it according to the picture. Your job is to fold it where the lines are so that the numbers are sorted - number 1 will be on the top, 2 under it,..., and the last one will be 8.

Modification: It is the same objective, just the numbers have changed.

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Folding Paper - solutionTrust me - it is possible. I have done it.

Slice the picture into 2 identical (Edit: symmetrical jagged cut) sections.

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2 Identical Sections - solution

Write the numbers from 1 to 8 into the squares, so that the squares with consecutive numbers do not touch (neither edges nor corners).

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Write Numbers - solution

The five figures shown below represent the appearance of a solid, opaque object as seen from five of its six sides. Each line shown depicts a side of the object that is perpendicular to the plane of this page. The object was constructed by gluing together a number of identical cubes so that at least one face of each added cube precisely and entirely covers and is everywhere contiguous with one face of a previous cube. Draw the sixth view of the object.

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Hoeflin's Object II. - solution

Where does the hole in second triangle come from (the partitions are the same)?

The same principle - moving the same parts - allows creating objects 64, 65 and 63 squares big. This geometric fallacy is also known as '64 = 65 Geometry Paradox'.

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64 = 65 Geometry Paradox - solutionIt looks like a triangle, because a thick line was used. Hypotenuse of the composite triangle is actually not a straight line – it is made of two lines. Forth cusps are where the arrows point (c9, l6).

The 64 = 65 paradox arises from the fact that the edges of the four pieces, which lie along the diagonal of the formed rectangle, do not coincide exactly in direction. This diagonal is not a straight segment line but a small lozenge (diamond-shaped figure), whose acute angle is

arctan 2/3 - arctan 3/8 = arctan 1/46which is less than 1 degree 15' . Only a very precise drawing can enable us to distinguish such a small angle. Using analytic geometry or trigonometry, we can easily prove that the area of the "hidden" lozenge is equal to that of a small square of the chessboard.

Slice the picture into 2 sections from which you could make a 8x8 square.

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8x8 Square - solution

Several identical cubes are fused together to form a solid object. Given the following five external views of such an object, draw the sixth external view. Clockwise or counterclockwise rotations of the sixth view are acceptable, but a mirror image (the sixth side as viewed from inside the solid) is not acceptable.

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Hoeflin's Object I. - solution

Slice the rectangle with a hole in its centre into 2 sections so that you could make a square 8x8 (without that hole in the centre).

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Lift - solution

The five images below represent five views (from five of its six sides) of a solid object. This object has been assembled by gluing together several identical small cubes so that at least one face of each small cube is totally adherent to a face of another contiguous small cube. Each black line shown represents a side of the object that is perpendicular to the plane of this page. Draw the sixth view of the object and calculate how many small cubes were used in the construction of the object, as well as its respective colors.

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Five Views - solution