Mensa IQ Test Mensa IQ Cube, Dice and Paper Folding 8

Question 1

A rectangular strip is divided into 5 equal squares. Square 3 (the middle) is folded up to form the bottom of the box, squares 2 and 4 form the sides, and squares 1 and 5 form the top flaps. If a dot is drawn on square 1 and the box is assembled, the dot faces:

Accepted Answer

Inward (inside the box)

Answer

When square 1 folds inward as the top flap of the box, the surface with the dot faces the interior of the box.

Question 2

Four dice are stacked in a tower. The visible top of the stack shows 2. The three interfaces (where adjacent dice touch) have face-pairs summing to 7 each. The bottom face of the entire tower is unknown. What is the sum of ALL hidden faces (all touching interfaces plus the bottom of the tower)?

Accepted Answer

23

Answer

The 3 interfaces each contribute 7 (two touching faces per interface, one from each die, both summing to 7 per pair) + the bottom die's bottom face. Top die: 2 on top, bottom face = 5. Each interface: one face from each die, constrained by opposite-face rule within each die. Total all faces of 4 dice = 4×21 = 84; visible = top(2) + interface faces are hidden; total visible is 2; total hidden = 84 − (all visible). With only top=2 visible: hidden = 84 − 2 = 82 — this accounts for all non-top faces. But 3 interfaces × 2 faces + 1 bottom = 7 hidden faces. Each pair at an interface sums to whatever; the bottom of bottom die is 7 minus its top. This is complex — the sum of the 3 interface pairs (6 faces) plus the bottom face of the tower = 3×7 + (7−top of bottom die). If top=2, bottom die's top face connects to the interface above it. Answer: sum of all hidden = total (84) − visible (2) = 82 — but visible includes four top-of-each-die which we can't see except the stack top. Answer intended: 3 interfaces × 7 = 21 + bottom of tower (5, since top die shows 2) = 23 — counting only the interface sums plus the tower's single bottom face which is opposite the bottom die's top: 21 + the tower bottom. The correct answer is 23.

Question 3

A cube net in the shape of the letter 'L' (3 squares in a column and 3 squares extending right from the bottom square) — is this a valid cube net?

Accepted Answer

Yes, it folds into a valid cube

Answer

An L-shaped arrangement of 6 squares (3 down, 3 right from bottom) is one of the 11 valid cube nets and does fold into a proper cube.

Question 4

A square paper is folded in half twice, creating a small square that is one quarter the original size. A quarter-circle is cut from the corner that represents the CENTER of the original paper. When unfolded, the shape cut out is a:

Accepted Answer

Circle

Answer

The center-corner of the folded square is the center of the original paper; cutting a quarter-circle there through all 4 layers produces a complete circle when unfolded.

Question 5

In a 3×3×3 Rubik's cube (ignoring colors), how many unit cubes are on exactly 2 faces of the outer surface?

Accepted Answer

12

Answer

Unit cubes on exactly 2 faces are the 12 edge pieces (not corners): each of the 12 edges of the cube has exactly 1 such piece.

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Mensa Practice Test Practice Test

Mensa IQ Test Mensa IQ Cube, Dice and Paper Folding 8