39 of 100: Stairway to Puzzlement

If the full cube has area 6, each side of it has area 1.

Top and bottom contribute 3 each to the area, as each square is full size. Total contribution 6. Left and right add 1 each, as $\frac12+\frac13+\frac16=1$ and the width is also 1. (Just think, it was 1 to start with, so moved around it is still 1.) Total added 2.

Front and back amount to area of $\frac12$ each, as the darkest rectangle is $\frac12$ high and 1 wide. From the back you see the $\frac12\times1$ rectangle of dark blue, from the front the three stacked to the same amount as shown in the figure. Total addition 1.

Total $6+2+1=\boxed9$ .

Any cut adds two new faces. The three pieces, separate, have a surface area of 6+2+2 = 10.

The overlapping parts after the joinings account for 2* $\frac{1}{6}$ and 2* $\frac{1}{3}$ .

$\frac{2}{6}$ + $\frac{2}{3}$ = $\frac{1}{3}$ + $\frac{2}{3}$ = 1, so the result is 10 - 1 = 9.

Steps:
1) Start with the full cube: Surface area = 6
2) Take the thickest piece off of the intermediate piece, uncovering 2 units. 6 + 2 = 8
3) Take the intermediate piece off of the thinnest piece, uncovering 2 more units. 8 + 2 = 10
4) Put the thickest and intermediate pieces together, covering back up $2 \times \frac{1}{3}$ units. $10 - \frac{2}{3}$
5) Put the intermediate and thinnest pieces together, covering back up $2 \times \frac{1}{6} = \frac{1}{3}$ units. $10 - \frac{2}{3} - \frac{1}{3} = \boxed{9}$

(1x6) + (1x4) - (1/6 + 1/3)x2 = 10-1 = 9

from the front view:1/6 +1/6 +2/3 =1 and the back side:1 and top & bottom:6 and left & right:2

so, the surface area of the new figure is= 1 + 1 + 6 + 2 = 9

It should be "area" in the second sentence...but it is obviously understood...just a typing error.😊

the front view: $\frac{1}{6}$ + $\frac{1}{6}$ + $\frac{2}{3}$ =1 the back side:1 top & bottom:6 left & right:2 1 + 1 + 6 + 2 = 9

Full breakdown of maths

1+1=2 ----- 2*3 = 6 This is for the top and bottom of each blue sections.

$\frac{1}{2}$ * 3 = 1.5 For the 3 darkest blue sides.

$\frac{1}{3}$ * 2 = 0.666r For the 2 medium blue sides.

$\frac{1}{6}$ * 3 = 0.5 For the 3 lightest blue sides.

1.5 + 0.667 + 0.5 = 2.666r

1/2 - 1/3 = 0.1666r For the front dark blue side that is partially blocked by the medium blue block.

1/3 - 1/6 = 0.1666r For the front medium blue side that is partially blocked by the lightest blue block.

0.1666r + 0.1666r = 0.333r

Final answer = 6 + 2.666r + 0.333r = 9

(Note: r stands for recurring)

Surface area = area which can be touched i.e., 6 faces in this structure

1. From front of the staircase = $\frac{1}{6}$ + $\frac{1}{6}$ + $\frac{1}{6}$ = $\frac{1}{2}$

2. Left of the staircase = $\frac{1}{6}$ + $\frac{1}{3}$ + $\frac{1}{2}$ = 1

3. Right of the staircase = $\frac{1}{6}$ + $\frac{1}{3}$ + $\frac{1}{2}$ = 1

4. Back of the staircase = $\frac{1}{2}$

5. Top of the staircase = 1 + 1 + 1 = 3

6. Bottom of the staircase = 1 + 1 + 1 = 3

Total surface area = $\frac{1}{2}$ + $\frac{1}{2}$ + 1 + 1 + 3 + 3 = 9

To keep the calculations down, think in terms of pairs: Front/Back, Left/Right, Top/Bottom. Then think of the simplest way to represent each face.

1. Surface area = 6 thus each square has area 1, and is 1x1.
2. Top/Bottom is simply 3 of the original square faces: 2x3 = 6
3. Left/Right are the three colors which add up to one square face: 2x1 = 2
4. Front/Back has potential for confusion when viewed from the front, so just look at the back, a half-square: 2x(1/2) = 1
5. Sum is 6+2+1 = 9

LaTeX: $\Large \text {the area}\; A_1 \; \text {is compound of 6 squares of the same size. }$

LaTeX: $\text {Let} \; \Large n \; \normalsize \text {the length of the cube side or the square side}$

LaTeX: $\Large A_1 = 6n^2 = 2n^2 + 4n^2 = 2n^2 + 4(\color{#3D99F6}{\frac{n^2}{2} + \frac{n^2}{3} + \frac{n^2}{6}})$

LaTex: $\Large \text {Let} \; (a = 2n^2), (\color{#3D99F6}{b = \frac{n^2}{2}}), (\color{#3D99F6}{c = \frac{n^2}{3}}\color{#333333}{) \; \text {and}} \; (\color{#3D99F6}{d = \frac{n^2}{6}})$

LaTex: $\text {We get : } \; \Large A_1 = a + 4 \color{#3D99F6} {(b + c + d)}$

LaTeX: $\Large \text {As you can see, the cube is split into 3 cuboids to represent a staircase}$

LaTeX: $\large \text { But the upper and lower areas don't change :} \; \Large a \normalsize \; \text {is multiplied by 3}$

LaTeX: $\large \text { Facing the staircase, 2 areas are half hidden :} \; \Large b \normalsize \; \text {and} \; \Large c$

LaTeX: $\Large b \; \normalsize \text {is partially hidden by} \; \Large c \; \normalsize \text {and} \Large \; c \; \normalsize \text {is partially hidden by} \; \Large d$

LaTex: $\text {We get : } \; \Large (b - c) + (c - d) + d = b$

LaTex: $\text {It was easy to guess. The area represents the back of the staircase, that is to say :} \; \Large b$

LaTex: $\text {The areas of the 2 sides of the staircase are : } \; \Large 2\color{#3D99F6} {(b + c + d)}$

LaTex: $\text {We have now a new area : } \;\Large A_2 = 3a + 2b + 2\color{#3D99F6} {(b + c + d)}$

LaTex: $\Large A_2 = 6n^2 + 2\frac{n^2}{2} + 2\color{#3D99F6}{n^2} \color{#333333}{= 9n^2 }$

LaTex: $\text {As :}\; \Large n = 1, \; \text {the new area is : } \; \boxed { \color{#D61F06} {\Large A_2 = 9 }}$

Separating the three cuboids, we have a total surface area of 2 + 4 * $\frac{1}{6}$ + 2 + 4 * $\frac{1}{3}$ + 2 + 4 * $\frac{1}{2}$ = 2 * 3 + 4 ( 1 ) = 10 . Then we take out the double counted areas which are 2 * $\frac{1}{6}$ + 2 * $\frac{1}{3}$ = 2 * $\frac{1}{2}$ = 1 . Hence, the required surface area = 9.

6+4(1)(1)-2(1/3)(1)-2(1/6)1=9

Orienting as if you are going to walk up the stairs we will look at the total areas facing in each direction.

Top and Bottom are easy, 3 squares of area 1, 3 square units each.

The Left and Right surfaces are just the original left and right sides of the cube redistributed so these are 1 square unit each.

The Back is a rectangle 1 unit wide and 1/2 unit high so that has area 1/2.

The three exposed front surfaces actually can be visualized as "covering" the back rectangle. The areas of the pieces sum to another 1/2.

Surface Area =3+3+1+1+1/2+1/2 =9

Original cube, Top+Bottom=2,.
New figure, Top+Bottom=6,.
Original cube, Left+Right=2,.
New figure, Left+Right=No change=2,.
Original cube, Front+Back=2,.
New figure, Back = 1/2,.
So, Front = Back = 1/2,.
New figure, Total = 6 + 2 +(1/2)+(1/2),.
= 9.