TLDR;
Alright, so this session is all about cubes and dice, focusing on cutting cubes, dice problems, and coloring cubes. We'll be looking at how to tackle these problems for the UPSC exam, including the types of questions you can expect and some handy shortcuts.
- Cube basics: vertices, edges, faces, and opposite faces.
- Cutting cubes: formulas for calculating pieces from cuts and vice versa.
- Problem-solving: strategies for maximizing or minimizing pieces with a fixed number of cuts.
Introduction to Cubes and Dice [18:02]
So, today we are going to discuss cubes and dice. The topic includes three subtopics: cutting of cubes, problems based on dice, and coloring of cubes and cuboids. In the UPSC 2024 exam, there was one question from cutting of cubes. In 2025, there were two statements, one related to cutting of cubes and the other related to coloring of cubes and cuboids.
Basic Properties of a Cube [19:40]
A cube is a 3D figure with length, breadth, and height. It has 8 corners (vertices), 12 edges, and 6 faces. Opposite faces are in three pairs: left-right, front-back, and top-bottom.
Cutting of Cube: One Axis [23:46]
In cutting of cube questions, you'll usually get the number of cuts and have to find the number of pieces, or vice versa. Remember, cuts are straight and go all the way through the cube. If you make one cut on one axis, you get two pieces. Two cuts give you three pieces, and so on. The formula here is: n cuts give you n + 1 pieces.
Cutting of Cube: Two and Three Axes [27:27]
When using two axes, like making one cut on the x-axis and one on the y-axis, you get four pieces. If you make one cut on one axis and two on another, you get six pieces. With three axes, one cut on each gives you eight pieces.
Formula for Cutting Cubes Along Three Axes [36:05]
If you make 'n' cuts along the x-axis, you get n + 1 pieces. Similarly, 'm' cuts on the y-axis give m + 1 pieces, and 'p' cuts on the z-axis give p + 1 pieces. The total number of pieces is (n + 1) * (m + 1) * (p + 1). So, just add one to each cut and multiply to get the total pieces.
Type 2: Fixed Number of Cuts [42:52]
Here, the examiner fixes the number of cuts but doesn't tell you how to distribute them across the axes. For a fixed number of cuts, you get the minimum pieces when you use only one axis. The more axes you use, the more pieces you get. For maximum pieces, try to distribute the cuts evenly across all three axes.
Minimum and Maximum Pieces with a Fixed Number of Cuts [48:15]
If you have to make 'n' cuts, the minimum number of pieces is n + 1 (all cuts on one axis). For the maximum, distribute the cuts as evenly as possible across the three axes.
Type 3: Non-Multiples of Three [55:07]
When the number of cuts isn't a multiple of three, you can't distribute them perfectly evenly. In this case, aim for minimum disparity. For example, if you have 10 cuts, distribute them as 3 + 3 + 4.
Chocolate Distribution Method [1:01:51]
Think of the cuts as chocolates to be distributed among three axes (or "friends"). Either distribute them evenly, or have a minimum difference in the distribution. If you have one extra chocolate, give it to your "favorite" axis. If you're short one, take one from the axis that can spare it the most.
Pieces to Cuts: One Axis [1:15:14]
Now, let's switch gears. If you want 10 pieces along one axis, you need 9 cuts. For 'n' pieces, you need n - 1 cuts.
Pieces to Cuts: Three Axes [1:16:28]
If you want 'n' pieces along the x-axis, 'm' pieces along the y-axis, and 'p' pieces along the z-axis, you need n - 1, m - 1, and p - 1 cuts, respectively. Add these cuts to get the total number of cuts required.
Observation: Pieces to Cuts [1:20:35]
If you use one axis, you need maximum cuts. As you increase the number of axes, the number of cuts decreases. To minimize the cuts, use the maximum number of axes (which is three).
Maximum and Minimum Cuts for a Fixed Number of Pieces [1:24:08]
For a fixed number of pieces, you need to find the maximum and minimum cuts. For maximum cuts, use only one axis. For minimum cuts, break the number of pieces into factors and use as many axes as possible.
Type 2: Pieces to Cuts - Clubbing Concept [1:28:45]
Sometimes, you'll have more than three factors. In this case, you need to "club" two factors together. When clubbing, prefer smaller multiples because they require fewer cuts.
Type 3: Pieces to Cuts - Perfect Cube Numbers [1:41:55]
If the number of pieces is a perfect cube, don't bother with clubbing. Just use n * n * n, where n is the cube root of the number.
GMA PYQ - Counting Triangles [1:44:48]
For 3D figures, manual counting is necessary. Consider both the front and back sides, using the bold and dotted lines as guides.
Number Series PYQ - Prime Number Difference [1:50:01]
In this number series question, the difference between consecutive numbers follows a prime number sequence.
Number Series PYQ - Alternate Series [1:51:28]
This question involves an alternate series where odd and even terms follow the same pattern.
Number Series PYQ - Identifying the Pattern [1:53:18]
This number series question involves multiplying with a fraction and then with a whole number.
Number Series PYQ - Odd Man Out [1:55:04]
This question asks to choose the group which is different from the others. The answer is D because it contains a composite number, while the others contain only prime numbers.
Number Series PYQ - Month Sequence [1:56:33]
This question involves a sequence of months where you subtract an increasing number from each month to get the next.
Number Series PYQ - Pythagorean Triplet [1:58:15]
This question is based on Pythagorean triplets.
Number Series Home Assignment - Difficult Series [1:59:25]
This is a difficult question because it is a small series. The pattern used for this question is further there will be difference 6, 18, 54.
Coding Decoding PYQ - Reverse Ranking [1:59:25]
In this question, the order of the letters in the English alphabet is reversed. Each letter represents the letter whose position it occupies.
Coding Decoding PYQ - Exact Same Ranking [2:07:40]
In this question, the pattern is they use exactly the same ranking.
Coding Decoding PYQ - Arranging Alphabetically in Reverse Order [2:09:56]
The letters of the word are arranged alphabetically in reverse order. The question asks how many positions of the letters will remain unchanged.
Coding Decoding PYQ - Pattern -1 +1 [2:13:43]
In this question, the pattern is -1 +1 -1 +1 -1 +1.
