__Dice__

__Dice__

Dice are cubes .They have six faces. Traditionally one to six dots are marked on each face and each face has different number of dots.

Dice can be standard or ordinary.

- Standard dice – Sum of numbers on each pair of opposite faces is 7. The face opposite to 6 is 1, the face opposite to 4 is 3 and the face opposite to five is 2.

Also none of the adjacent numbers will add upto7.

- Ordinary dice- Not all pairs of numbers on opposite faces add up to 7 or at least one set of visible adjacent numbers add up to 7 .

** Logic 1:** If a number appears twice and all other numbers once in two throws then number which is not visible is opposite to the number appearing twice.

**Example 1:** A dice is rolled twice, and the positions shown below. Which number will be opposite 2?

a) 5 b) 6 c) 1 d) 4

**Solution:** 5,4,6 and 3 are neighbours of 1 .The only remaining number is 2 .Therefore, the face opposite to 1 is 2 or we can say that the number opposite to 2 is ** 1**.

** Logic 2 : **A dice is rolled twice – 2 numbers appear twice, only the third number is different, then the 3

^{rd}number (nonrepeating numbers) in both the positions will be opposite each other.

** Explanation:** Opposite faces of the numbers appearing twice will not be visible in any of the positions that means four numbers -numbers that appear twice and their opposite faces i.e., numbers which are not visible in any of the positions, are already exhausted. Hence only two non-repeating visible numbers remain, and they will be opposite each other.

**Example 2: **Two positions of a dice are as shown below. Which number will be opposite 4.

a)6 b) 3 c) 1 d) 5

**Solution:**

Four numbers are visible . Two numbers which are not visible (2 and 3) are opposite 1 and 5 as 1 and 5 are visible in both the positions.

Therefore, number opposite 4 must be ** 6** as all other choices are exhausted.

** Logic 3** : If a symbol(number) appears twice at same position then symbols appearing on same side in two different positions of the die will be opposite to each other.

Visualisation plays an especially important role here.

**Example 3:** Two positions of a dice are shown below. Which number will appear opposite 5.

a)6 b) 3 c) 1 d) 5

**Solution:**

By Logic 3:

Number opposite 5 will be 2 as appears on same side(top) twice. Hence numbers on same side (front in both positions) of die will be opposite each other.

By Visualisation:

Rotate 2nd dice anticlockwise 3 goes behind, 2 comes to right, number in front will be 6 (refer 1st position)

Again, rotate the 2nd dice anticlockwise 3 comes to left, 2 goes behind, 6 comes to right and 5 comes to front as shown in 1st position.

**Example 4:** A dice thrown twice appears as shown below. Symbols engraved on the faces of dice are @,&,#,%,* and $.Which symbol is at the bottom when @ is on top?

a)& b) # c) % d) *

**Solution:** a) &

Rotate 2^{nd} die clockwise such that “&” goes at the bottom , # goes to right and @ comes to the top ( refer 1^{st} position).

Therefore, ** &** is at the bottom when @ is on top.

__Nets of Cubes and dice __

A Net is a flat shape which forms a 3-D shape on folding.

A 3-D shape when opened and flattened will look like one of its nets.

Some of the common nets of the dice are shown below.

** Logic 4**: When the nets are folded to form dice, symbols appearing at alternate places of net will appear on opposite

faces of the die.

For nets a) and b) faces with same symbols are opposite to each other.

For net c) 1 is opposite 3, 4 is opposite 6, and remaining two numbers 2 and 5 are opposite each other.

For net d) 6 dots are opposite 2 dots. 4 dots are opposite 3 dots and hence 5 dots will be opposite 1 dot.

**Example 5:** Choose the cube that can be formed with the below net.

**Solution:** b)

**Explanation:**

2 and % should be opposite each other. So a) is not correct.

Similarly, c) and d) are not correct .

**Example 6:** Which of the dice can be constructed with the given net?

**Solution:** d)

From the net- 1 is opposite 3. So, b) is wrong.

4 is opposite 6. So, c) is wrong.

5 is opposite 2. So, a) is wrong.