Cube Stacking

Calculate the 5 total faces of the 10cm cube. The top face covers all the top faces of the other cubes. So for the other cubes. Calculate 4 faces. 4+16+36+64+100+144+196+256+324+500= 1640

Lets do it step by step:

Step 1 Calculate the area of one side of each cube: For the 1 cm cube, the area of one side is = 1cm^{2} For the 2 cm cube, the area of one side is = 4cm^{2} For the 3 cm cube, the area of one side is = 9cm^{2} For the 4 cm cube, the area of one side is = 16cm^{2} For the 5 cm cube, the area of one side is = 25cm^{2} For the 6 cm cube, the area of one side is = 36cm^{2} For the 7 cm cube, the area of one side is = 49cm^{2} For the 8 cm cube, the area of one side is = 64cm^{2} For the 9 cm cube, the area of one side is = 81cm^{2} For the 10 cm cube, the area of one side is = 100cm^{2}

Step 2 Calculate area of four sides which are prependicular to the table surface(unless you mess up with gravity or something) For the 1 cm cube, the area of four sides is = 4cm^{2} For the 2 cm cube, the area of four sides is = 16cm^{2} For the 3 cm cube, the area of four sides is = 36cm^{2} For the 4 cm cube, the area of four sides is = 64cm^{2} For the 5 cm cube, the area of four sides is = 100cm^{2} For the 6 cm cube, the area of four sides is = 144cm^{2} For the 7 cm cube, the area of four sides is = 196cm^{2} For the 8 cm cube, the area of four sides is = 256cm^{2} For the 9 cm cube, the area of four sides is = 324cm^{2} For the 10 cm cube, the area of four sides is = 400cm^{2} Total=1540cm^{2}

Step 3 Calculate the area visible from the top The total area of top sides will be 100cm^2 since the smaller cube sits on the bigger cube and hides the area that it is covering. Just think how it will look from the top and you will understand (I hope!)

Step 4 Sum the areas 1540cm^{2} +100cm^{2} = 1640cm^{2}

BINGO!!!

Thats the answer. Its a rather confusing question since its so long but one can do it with logical thinking and imagination!!!

Well I have got yet an another approach...

the first cube will have all five faces visible = 5 $cm^{2}$

the second one will have 4(4) + (4-1) =19 $cm^{2}$ (since the upper cube has covered the area of one side on the cube below it...)

The same is done with all the remaining cubes :

3) 4(9) + (9-4) =41 $cm^{2}$

4) 4(16) + (16-9) =71 $cm^{2}$

5) 4(25) + (25-16) =109 $cm^{2}$

6) 4(36) + (36-25) =155 $cm^{2}$

7) 4(49) + (49-36) =209 $cm^{2}$

8) 4(64) + (64-49) =271 $cm^{2}$

9) 4(81) + (81-64) =341 $cm^{2}$

10) 4(100) + (100-81) =419 $cm^{2}$

adding all these we get $\boxed{1640}$