The six faces of a wooden cube of side 6 cm are labelled A, B, C, D, E and F respectively. Three of these faces A, B, and C are each adjacent to the other two, and are painted red. The other three faces are not painted. Then, the wooden cube is neatly cut into 216 little cubes of equal size. How many of the little cubes have no sides painted?
- 125
- 135
- 91
- 108
- 100
The Correct Option is A
Solution and Explanation
Step 1: Break down the problem
The cube has side length = \(6 \, \text{cm}\). When cut into smaller cubes of side \(1 \, \text{cm}\), the total number of small cubes is: \[ 6^3 = 216 \]
Step 2: Painted vs unpainted sides
Faces A, B, C are painted. Importantly, these three faces are mutually adjacent (they meet at one corner). Faces D, E, F are unpainted.
Step 3: Condition for unpainted small cubes
A little cube has no side painted if it lies entirely within the "interior" of the big cube, i.e., away from the 3 painted faces.
Step 4: Effective interior cube
Since A, B, and C meet at one corner, any cube on those faces will be painted. To avoid them, we must exclude 1 layer from each of the painted faces. So the "completely safe" unpainted inner block has side: \[ (6 - 1)^3 = 5^3 = 125 \]
Step 5: Conclusion
Therefore, the number of little cubes with no painted sides is: \[ \boxed{125} \]
Top Questions on Geometry
- If \[ \lim_{x \to 0} \frac{e^{a(x-1)} + 2\cos(bx) + e^{-x}(c - 1)}{x \cos x - \ln(1 + x)} = 2, \] Then the value of \( a^2 + b^2 + c^2 \) is:
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- If \( \cos(\alpha + \beta) = -\frac{1}{10} \) and \( \sin(\alpha - \beta) = \frac{3}{8} \), where \[ 0<\alpha<\frac{\pi}{3} \quad \text{and} \quad 0<\beta<\frac{\pi}{4}, \] and \[ \tan(2\alpha) = \frac{3\left(1 - \sqrt{55}\right)}{\sqrt{11} \left(s + \sqrt{5}\right)}, \] and \( r, s \in \mathbb{N} \), then \( r^2 + s \) is:
- If \( O \) is the vertex of the parabola \( x^2 = 4y \), \( Q \) is a point on the parabola. If \( C \) is the locus of a point which divides \( OQ \) in the ratio \( 2:3 \), then the equation of the chord of \( C \) which is bisected at the point \( (1,2) \) is:
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Questions Asked in XAT exam
- An artificial swimming pool is circular and water is flowing clockwise at a speed of 3 km/hr. Two swimmers Ayub and Rana start from diametrically opposite points, Ayub in the anticlockwise direction and Rana in the clockwise direction. They first meet at a point such that Ayub has covered 60 m by that time. Their second meeting point is such that Rana has covered another 180 m after the first meeting point. If the speed of Rana in still water is 3 km/hr, find the time taken by Ayub to cover one full circle in the clockwise direction.
- XAT - 2026
- Time and Work
- Arrange the following generations from the oldest to the youngest:
1. Generation X
2. Silent Generation
3. Generation Alpha
4. Baby Boomers
5. Millennials
- XAT - 2026
- Current Affairs
- Read the following scenario and answer the THREE questions that follow.
Brijbhushan, a microfinancier, lends money at the rate of Rs.10 per square meter to small farmers at a village. He charges an annual interest rate of 10%. All the farming plots in that village are rectangular, with areas varying between a minimum of 1000 square meters and a maximum of 10,000 square meters.
This year, Brijbhushan has lent money only to five farmers: Aditya, Binod, Chhuttan, Dabloo and Govind. The perimeter of Chhuttan’s plot is 250 meters, with the length and width being at a ratio of 4:1. Aditya’s plot has an area three times the area of Govind’s plot. The area of Aditya’s plot is also the average of the areas of Govind’s plot and Dabloo’s plot. The plots belonging to Aditya, Binod and Dabloo are of the same width, but of different lengths. Moreover, the length of Binod’s plot is the sum of the lengths of Aditya’s plot and Dabloo’s plot.- XAT - 2026
- Mensuration
Light Chemicals is an industrial paint supplier with presence in three locations: Mumbai, Hyderabad and Bengaluru. The sunburst chart below shows the distribution of the number of employees of different departments of Light Chemicals. There are four departments: Finance, IT, HR and Sales. The employees are deployed in four ranks: junior, mid, senior and executive. The chart shows four levels: location, department, rank and gender (M: male, F: female). At every level, the number of employees at a location/department/rank/gender are proportional to the corresponding area of the region represented in the chart.
Due to some issues with the software, the data on junior female employees have gone missing. Notice that there are junior female employees in Mumbai HR, Sales and IT departments, Hyderabad HR department, and Bengaluru IT and Finance departments. The corresponding missing numbers are marked u, v, w, x, y and z in the diagram, respectively.
It is also known that:
a) Light Chemicals has a total of 210 junior employees.
b) Light Chemicals has a total of 146 employees in the IT department.
c) Light Chemicals has a total of 777 employees in the Hyderabad office.
d) In the Mumbai office, the number of female employees is 55.
- XAT - 2026
- Pie Charts
An investment company, Win Lose, recruit's employees to trade in the share market. For newcomers, they have a one-year probation period. During this period, the employees are given Rs. 1 lakh per month to invest the way they see fit. They are evaluated at the end of every month, using the following criteria:
1. If the total loss in any span of three consecutive months exceeds Rs. 20,000, their services are terminated at the end of that 3-month period,
2. If the total loss in any span of six consecutive months exceeds Rs. 10,000, their services are terminated at the end of that 6-month period.
Further, at the end of the 12-month probation period, if there are losses on their overall investment, their services are terminated.
Ratan, Shri, Tamal and Upanshu started working for Win Lose in January. Ratan was terminated after 4 months, Shri was terminated after 7 months, Tamal was terminated after 10 months, while Upanshu was not terminated even after 12 months. The table below, partially, lists their monthly profits (in Rs. ‘000’) over the 12-month period, where x, y and z are masked information.
Note:
• A negative profit value indicates a loss.
• The value in any cell is an integer.
Illustration: As Upanshu is continuing after March, that means his total profit during January-March (2z +2z +0) ≥
Rs.20,000. Similarly, as he is continuing after June, his total profit during January − June ≥
Rs.10,000, as well as his total profit during April-June ≥ Rs.10,000.
- XAT - 2026
- Logical and Analytical Reasoning Skills