logical reasoning - cube problem

read this on pagalguy ... nice one !!

A cube is given with an edge of unit ‘N’. It is painted on all faces. It is cut into smaller cubes of edge of unit ‘n’. How many cubes will have ‘x’ faces painted?

In these types of questions, the first thing that we need to figure out is the number of smaller cubes. For this, we look at one particular edge of the big cube and figure out how many smaller cubes can fit into this. It will be N/n. So, the number of smaller cubes will be (N/n)³

A cube has 6 faces and none of the smaller cubes will have all faces painted. As a matter of fact, none of the smaller cubes will have even 5 or 4 faces painted. The maximum number of faces, which will be painted on a smaller cube, will be 3. This will happen only in the case of the smaller cubes that emerge from the corners of the big cube.

So, number of smaller cubes with 3 faces painted = 8 (Always)

For 2 faces to be painted, we will have to consider the smaller cubes that emerge from the edges of the big cube (leaving out the corners). So, the smaller cubes on every edge will be (N-2n)/n. There are 12 edges in a cube.

So, number of smaller cubes with 2 faces painted = 12 * (N-2n)/n

For 1 face to be painted, we will have to consider the smaller cubes that emerge from the face of the big cube (leaving out the corners and the edges). So, the smaller cubes on every face will be [(N-2n)/n]². There are 6 faces in a cube.

So, number of smaller cubes with 1 face painted = 6 x [(N-2n)/n]²

For no face to be painted, we will have to consider the smaller cubes that emerge from the inside of the big cube (leaving out the outer surface which was painted). Imagine this as taking a knife and cutting a slice of width ‘n’ from every face of the cube. You will be left with a smaller cube with an edge of ‘N-2n’. Number of smaller cubes that you can make from the resulting cube is [(N-2n)/n]³

So, number of smaller cubes with 0 face painted = [(N-2n)/n]³

Let us take an example to elucidate this type of problem.

Example,

A painted cube is given with an edge of 15 cm. Smaller cubes are cut out from it with an edge of 3 cm each. How many cubes will have 3 faces painted, 2 faces painted, 1 face painted and no face painted.

Solution,

Total number of smaller cubes = (15/5)³ = 125

3 faces painted = 8 cubes.

2 faces painted: Consider an edge of size 15 cm. We have removed the corners that take away 3 cm from each corner of the edge. Now our edge is of 9 cm. 3 cubes of 3 cm each can come from it. There are 12 edges. So, there will be 3 * 12 = 36 cubes.

1 face painted: Consider a face. If we have removed 3 cm from each edge of the face, we will be left with a square of side 9 cm or area 81 sq cm. There can be 9 smaller squares that can be formed on that face with an area of 9 sq cm each. These 9 will be the cubes which will have 1 face painted. There are 6 faces. So, there will be 9 * 6 = 54 cubes.

No face painted: Cut slices of 3 cm each from each face of the cube. We will be left with a smaller cube of edge 9 cm. Number of smaller cubes that can be formed from it is (9/3)³ = 27. So, 27 cubes will have no faces painted.

You can use this to verify the formulas above and also note that 8 + 36 + 54 + 27 = 125. This means that there is no need to find out all four using the formula, just find any three of them and the other would emerge by using the total. In an exam, this might save you some valuable time.

Remembrance

Independence day just went past. In my patriotic fervor, i request you all to maintain 2 minutes silence and remember all those who struggled against the British ..... Dhoni, Tendulkar, Sehwag.....

Philosophy

If she vaunts herself, I abase her,
If she abases herself, I vaunt her;
And contradict her always,
Until she comprehends
That she is an incomprehensible monster.

Aamir & Anand !!

On winning the World Chess Championships in 2000, Anand was extensively felicitated across India. I happened to visit one such function organized, in his honor, at Bandra. Aamir Khan was the guest of honor along with Sunil Dutt, who was present as the felicitation was in his constituency . Aamir Khan is reputed to be a brilliant chess player. Anand analysed one of his games on a demo board for the benefit of the ecstatic crowd. All the time Aamir was standing and listening to each and every word of Anand's wisdom with attention. I clicked this photo just before the game analysis started.

Sunil Dutt, who was using a walking stick to support himself, also stood for the entire length of the game analysis. I presume this to be a mark of repect for the genius of Anand.

Anand was to be felicitated on the same day by the Maharashtra Govt. too (in the evening). The government felicitation was a joke. Anand was presented a cheque of Rs. 500000. Small change for Anand. He donated the amount then and there to the Prime Minister's earthquake relief fund.

War Memorial in Dombivli

One of the places that i visit often in Dombivli is the "War Memorial" in honor of Captain Vinay Kumar Sachan, who died in kashmir in the line of duty while in his early 20's. He was a resident of Dombivli. Kalyan Dombivli Municipal Corporation made a great effort in creating a Memorial in his honor (at the MIDC ground) and getting a decommissioned tank from the Indian Army to be placed at this memorial. Adjacent to the memorial is a small 'war museum of sorts', where personas of some martyrs are displayed in metal and art . I took some snaps there, which are seen below !

Captain Vinay Kumar Sachan

The Army tank at Captain Vinay's memorial

Depiction of Indian Army's victory in the Kargil War (atop Tiger Hill)

"Wagah Border Gate" ...now in Dombivli !

I had actually visited Wagah Border many times at the height of Indo- Pak tensions years ago. When the march happens, people from India sit on one side. People from Pakistan sit on the other. The gate between India and Pakistan is thrown open and a small military ceremony happens. It was no less than war at that time even at his march ! When the Indian soldiers stomped, Indians used to cheer and pakistanis used to boo and vice versa. Officers on both sides tried hard to assuage the feelings !!

Pakistan resigning during the creation of Bangladesh in 1971. 90000 Pakistani soldiers laid down their arms and surrendered to the Indian Army. Under the Geneva War Convention, "Prisoners of War" can't be ill-treated. They were sent back to Pakistan with individual dignity upholded . Pakistan's dignity though was severely wounded.

Second Lieutenant Rama Raghoba Rane

Lance Naik Karam Singh

Naik Jadunath Singh

Cross- View

Field Marshal Sam Maneckshaw, a legend in India

This is one of those places where i find absolute peace. A very small gallery, but a huge step for Dombivli. When people are busy building malls, residential complexes and commercial establishments, here is one sterling example of the cultural sensitivities of Dombivli.

Saw Shammi Kapoor

Went to Breach Candy hospital today for a battery of tests . Saw Shammi Kapoor (the bollywood superstar of yester years !) coming in a wheel chair...He has thinned down a lot...probably due to illness.

I feel so happy to have seen him in person! As he came in i felt i knew him but wasn't able to place him...as i walked past him, it struck me and i became standstill. I turned back and saw him till he got into a lift :)

Obituary - Mr S.S.Natu

At 1.30 am, i heard a car starting. I woke up to see whether any theft was being attempted. What i saw was my neighbour's daughter getting into the car (after closing the main gate). I immediately knew that her dad, who was hospitalized, must have had some complication. He took his last breath today . His age must be around 75.

We have heard of lots of rags to riches stories. He lived through one, being the protagonist himself. From a penniless child to a multi - millionaire, his life has been an inspiration to the people around here. Many have sought his blessing while starting new ventures. I remember even an autowala's family taking his blessings while their son was embarking his life as an auto owner & driver. Yet few people outside Dombivli know him or his journey, for, he always liked to keep a low profile.

Normally when we think of any person, we always get a mixed baggage of feelings. There are some good impressions and then there are some bad impressions. However Mr. Natu is a man who has always evoked only positive feelings. It is amazing. But this is what comes out of a life lived to the fullest.

After my first year of MBA in 2008, we chanced to meet on the stairs. While moving down, he gave me two points of his business values, which have become a part of my own consciousness.

1. Always think big - He said that he was always amazed at how people think small and live small, worrying incessantly about EMI's, rejected loan applications,security etc . They never get out of the trap. So always think big. Don't live a hand-to-mouth existance.

2. Keep a very low profile and build your inner strength- This is one value which i espouse myself. He said we need to keep a low profile. He himself kept such low a profile that even his relatives/neighbours didn't know about his meteroic rise. Also people won't feel threatened and so they won't try to bring us down. While maintaining a low profile, we need to grow strong from within.

A life so successful is unfortunately not documented (to my knowledge). His daughters who run & grow the businesses now may take time out some day to pen his life. And even that will inspire many. As in life, so after.

May his soul rest in peace.

Amen