noteswale
Noteclerical

Dice / cube

Lesson 31 of 113

1. Ye topic hai kya?

Cube ki 6 faces — dice par number/dot, ya rang se paint karke chhote cubes. 2D ulti figure yahan nahi (Shapes / mirror).

Paper kya maangta

Kaun si face opposite, do photos same dice ki, kaun sa net fold hoga, n×n×n kaatne ke baad kitni faces painted.

Pehle lock

Ek view mein teen faces dikhti hain — wo teen ek hi kone par milti hain. Aapas mein koi opposite nahi.

2. Cube — 6, 12, 8

6 faces · 12 edges · 8 corners
Har face ke 4 padosi (edge se jude) aur 1 opposite (kabhi saath nahi dikhti)
Jo do faces ek hi picture mein dikh rahi hain, wo opposite nahi ho sakti. Opposite hamesha chhipi rehti hai us view mein.

3. Ordinary dice — tabhi 7

Kuch sawaal likhte hain “ordinary / standard dice” ya “opposite faces ka sum 7”.

Tabhi: 1 ↔ 6 · 2 ↔ 5 · 3 ↔ 4
Indian paper ke zyada dice ordinary nahi. Sum 7 tabhi use karo jab sawaal khud bole. Warna neighbour-list se nikaalo.

4. Do positions — yahi asli method

Same dice, do photos. Figure mein teen faces: saamne / upar / daayan.

Do common faces dono photos mein → jo teesri bachi, wo aapas mein opposite.

Ek common face X → dono photos ki baaki faces X ke padosi. Jo number 1–6 mein dikha hi nahi, wahi X ka opposite.
X ke 4 padosi mil gaye to sixth number = opposite. Agar 4 poore na milen, doosra rule (do-common) dekho — galat pair mat jodna.

5. Net — khula cube

6 squares ka pattern jo fold hoke cube bane. Seedha 4 se zyada squares ek line mein = fold nahi hota (5–6 wali strip galat).

Seedhi line mein ek square chhod kar jo do hain, fold ke baad opposite:
3-strip A–B–C → A opposite C
4-strip A–B–C–D → A opposite C, B opposite D
2×3 wala rectangle net ban sakta hai — “6 chipke hain isliye galat” mat sochna.

6. Painted cube, phir n×n×n kaato

Bada cube 6 taraf paint, phir chhote unit cubes. Poora n³.

Painted facesFormula3×3×34×4×4
3 (konas)8 hamesha (n≥2)88
2 (edges, kona chhod)12(n−2)1224
1 (face ke beech)6(n−2)²624
0 (andar)(n−2)³18
2×2×2: n−2=0. Sirf 8 cubes, sab 3-face painted. 2-face / 1-face / andar = 0.

7. Sawal — basic se pro

Q3–Q4, Q7 ek hi cube pe chalenge: saamne 1, peeche 2, upar 3, neeche 4, daayan 5, baayan 6. Opposite: 1–2, 3–4, 5–6. Ordinary nahi (1+2=3, 7 nahi).

Q1 · Basic

Ek view mein 2, 3, 5 dikh rahe hain (teen kone par milte hain). 2 ka opposite nahi ho sakta?

  1. 1
  2. 3
  3. 4
  4. 6

Solution

  1. Step 1

    Jo saath dikhe, wo opposite nahi. 2 ke saath 3 aur 5 hain.

  2. Step 2

    3 definitely nahi. 1, 4, 6 abhi possible — itna data unhe kaatne ko kaafi nahi.

AnswerB · 3
Q2 · Basic

Ordinary dice. 2 ke opposite kaun si face?

  1. 3
  2. 4
  3. 5
  4. 6

Solution

  1. Step 1

    Ordinary = opposite sum 7. 2 + 5 = 7.

AnswerC · 5
Q3 · Medium

Same dice, do positions. 1 ke opposite?

Position I saamne 1 · upar 3 · daayan 5
Position II saamne 5 · upar 3 · daayan 2
  1. 3
  2. 5
  3. 2
  4. 6

Solution

  1. Step 1

    I: 1, 3, 5. II: 5, 3, 2. Common do faces: 3 aur 5.

  2. Step 2

    Jo teesri bachi: I mein 1, II mein 2. Do-common rule: 1 opposite 2.

  3. Step 3

    3 aur 5 dono views mein hain — wo 1 ke opposite nahi (padosi).

AnswerC · 2
Q4 · Medium

Wahi dice. Position I wahi (1, 3, 5). Position II neeche. 3 ke opposite?

Position II saamne 2 · upar 3 · daayan 6
  1. 1
  2. 2
  3. 4
  4. 5

Solution

  1. Step 1

    I: 1, 3, 5. II: 2, 3, 6. Common sirf 3.

  2. Step 2

    Dikh numbers: 1, 2, 3, 5, 6. Jo nahi dikha: 4.

  3. Step 3

    Ek-common rule: jo sixth bacha, wahi common ka opposite. 3 ↔ 4. (1 aur 5, 3 ke padosi the — opposite nahi.)

AnswerC · 4
Q5 · Basic

3×3×3 cube, 6 taraf paint, phir 27 chhote. Kitne cubes ki exactly 2 faces painted?

  1. 6
  2. 8
  3. 12
  4. 27

Solution

  1. Step 1

    2-face = 12 edges × (n−2) = 12 × 1 = 12. (Har edge ke do konas 3-face wale hain, beech wala ek 2-face.)

AnswerC · 12
Q6 · Medium

4×4×4, 6 taraf paint, kaato. Kitne chhote cubes bina paint?

  1. 1
  2. 4
  3. 8
  4. 16

Solution

  1. Step 1

    Andar wala cube (n−2)³ = 2³ = 8.

  2. Step 2

    Check: kul 64. 3-face 8 + 2-face 24 + 1-face 24 + 0-face 8 = 64.

AnswerC · 8
Q7 · Medium

Yeh net fold hoke cube. 6 ke opposite?

3
6
1
5
4
2
  1. 1
  2. 2
  3. 4
  4. 5

Solution

  1. Step 1

    Row 6–1–5 teen strip: 6 opposite 5. (Beech 1 dono se judi, fold par 6 aur 5 do taraf.)

  2. Step 2

    Column 3–1–4–2: 3 opposite 4, 1 opposite 2. 6 ka opposite 1 nahi — 1 padosi hai.

AnswerD · 5
Q8 · Basic

Ordinary dice. 4 neeche hai. Upar kaun si?

  1. 2
  2. 3
  3. 5
  4. 6

Solution

  1. Step 1

    Neeche-upar opposite. 4 + 3 = 7. Upar = 3.

AnswerB · 3
Q9 · Medium

Kaun sa net cube nahi banata?

A
B
C
D

Solution

  1. Step 1

    C mein seedhi 5 squares. Cube ki belt 4 faces ki hoti hai — 5-strip fold hoke overlap. Galat net.

  2. Step 2

    A Latin-cross (6) theek. B 2×3 rectangle theek — trap. D plus+ek theek.

AnswerC · 5-strip
Q10 · Pro

Ordinary dice. Kaun sa view impossible?

A 1, 2, 3
B 1, 3, 5
C 2, 3, 5
D 1, 2, 4

Solution

  1. Step 1

    Ordinary: 2 ↔ 5, 1 ↔ 6, 3 ↔ 4. Jo pair ek view mein saath aaye, wo galat.

  2. Step 2

    C: 2 aur 5 dono dikh rahe — opposite pair visible. Impossible.

  3. Step 3

    A: 1,2,3 — koi opposite pair nahi, kone par mil sakte. B: 1,3,5 theek. D: 1,2,4 theek (3↔4 isliye 4 padosi hai 1,2 ki).

AnswerC · 2 aur 5 saath

Practice

Pehle khud, phir neeche kholo.

12345 678910
P1

View mein 1, 4, 6 dikhte hain. 1 ka opposite nahi ho sakta?

  1. 2
  2. 3
  3. 4
  4. 5

Solution

  1. Step 1

    1 ke saath 4 dikha. Opposite 4 nahi. Baaki options abhi possible.

AnswerC · 4
P2

Ordinary dice. 6 ke opposite?

  1. 1
  2. 2
  3. 3
  4. 5

Solution

  1. Step 1

    6 + 1 = 7.

AnswerA · 1
P3

Do positions. 6 ke opposite?

I saamne 6 · upar 4 · daayan 1
II saamne 1 · upar 4 · daayan 3
  1. 1
  2. 3
  3. 4
  4. 5

Solution

  1. Step 1

    I: 6,4,1. II: 1,4,3. Common do: 4 aur 1.

  2. Step 2

    Teesri: 6 aur 3. 6 opposite 3.

AnswerB · 3
P4

P3 wala dice. Position I (6, 4, 1) aur yeh II. 4 ke opposite?

II saamne 3 · upar 4 · daayan 2
  1. 1
  2. 2
  3. 5
  4. 6

Solution

  1. Step 1

    I: 6,4,1. II: 3,4,2. Common sirf 4. Dikh: 1,2,3,4,6. Bacha 5.

  2. Step 2

    4 opposite 5.

AnswerC · 5
P5

3×3×3, 6 taraf paint. Kitne cubes par paint bilkul nahi?

  1. 0
  2. 1
  3. 6
  4. 8

Solution

  1. Step 1

    (3−2)³ = 1 — bilkul beech wala.

AnswerB · 1
P6

5×5×5, 6 taraf paint. Exactly 2 faces painted?

  1. 12
  2. 24
  3. 36
  4. 48

Solution

  1. Step 1

    12(n−2) = 12×3 = 36.

AnswerC · 36
P7

Is net mein 2 ke opposite?

A
B
2
C
D
E
  1. A
  2. B
  3. C
  4. D

Solution

  1. Step 1

    Column A–2–E: 3-strip, A opposite E — poocha 2 ka nahi.

  2. Step 2

    Row B–2–C–D: 4-strip. B opposite C, 2 opposite D.

AnswerD · D
P8

Ordinary dice. 1 upar. Neeche?

  1. 2
  2. 4
  3. 5
  4. 6

Solution

  1. Step 1

    Upar-neeche opposite. 1 ↔ 6.

AnswerD · 6
P9

2×2×2 cube, 6 taraf paint, 8 pieces. Exactly 2 faces painted kitne?

  1. 0
  2. 4
  3. 8
  4. 12

Solution

  1. Step 1

    n−2=0. 12×0=0. Har chhota cube ek kona hai — 3 faces painted, 2-face koi nahi.

AnswerA · 0
P10

P3–P4 wale dice pe opposite 6–3, 4–5, 1–2. Kaun sa view impossible?

  1. 6, 4, 1 saath
  2. 6, 3, 4 saath
  3. 3, 4, 2 saath
  4. 6, 4, 2 saath

Solution

  1. Step 1

    B: 6 aur 3 opposite, ek view mein saath — impossible.

  2. Step 2

    A Position I tha. C P4-II tha. D: 6,4,2 — 1↔2 isliye 2 padosi ho sakta 6,4 ka (baayan-neeche-peeche wala kona nahi, lekin 6 ke padosi 4,5,1,2). 4 aur 2 dono 6 ke padosi; 4 ke padosi 6,1,3,2. Teen 6,4,2 milte hain. Theek.

AnswerB · 6 aur 3 saath

← Index · Age

Agle topic: Water image.