CPhill

If the product of all 6 rolls is, odd...EVERY roll must be odd, since any even roll - or multiple even rolls -  in the 6  will  result in an even product

We have a (1/2)  probability of an odd roll on each roll....so....the probability of  an odd product on  6  rolls  is :

(1/2)^6   =  1 / 64

ΔABE  ~ Δ BEC   which implies that

AB/ BE = BE / EC

And  ΔABE   is a 30 - 60 - 90 right triangle....AE  = 24  and BE  = 1/2 of this  = 12

So

24 / 12  = 12 / EC

12 /6  = 12 / EC

EC   =  CE  = 6

Thanks, azsun...here's one more way.....

We have that   6 + b + c  = 18   ⇒   c = [18 - 6  - b]  =  [12  - b]

So...using the  Pythagorean Theorem

6^2  + b^2  = (12 - b)^2

36 + b^2  = 144 - 24b + b^2

24b - 108  = 0

24b  = 108

2b  = 9

b  = 9/2  = 4.5

So...c  =  12  - 4.5  =  7.5

Mmmm...I assumed the second way....but I see your point, Guest  !!!

Let's assume your first way is the  correct one...so we have

1 2 3 4 5 6 7 8 9 10 11 12 13

12 11 10  9  8  7  6  5 4 3 2 1

2 3 4 5 6 7 8 9 10 11 12 13

12 11 10  9  8  7  6  5  4 3  2  1

So...the first  row contains 13 total digits...and every othe row will contain 12 digits

Subtracting 13 from 5000 we get 4987  digits remaiining

So   4987 / 12 = 415 * 12 + 7

So....this will be the 7 digit on the 416th row after the first one

Which will be the row  2 3 4 5 6 7 8 9 10 11 12 13

So...the 5000th digit will still   = 8  

Just wan tto try this one myself....!!!

The triangle will be a45-45-90 right triangle

Call the length of its legs, x

So...the area of the black triangle is  (1/2) (product of the leg lengths)  = (1/2)x^2

We also have two  "white" rectangles  each with an area  of  x ( √6 - x  )  = 2x (√6 - x)

And we have a "white" square  with an area  of (√6 - x)^2

Since the areas are equal...we have that

(1/2)x^2  = 2x(√6 - x)  + (√6 - x)^2

(1/2)x^2  = 2√6x - 2x^2  + 6  - 2√6x + x^2

(1/2)x^2  = -x^2 + 6

(3/.2)x^2  - 6   = 0

3x^2 - 12  = 0

3x^2  = 12

x^2  = 4

x =2

So...when  A  is folded over.....it will  be √ [ 2^2 + 2^2 ] = √8  =  2√2 cm  from it's original position

Note that we  have a  cycle of  26 digits - 13 up and 13  down

So

5000 mod 26  =  8

So 8 will be the 5000th digit

Proof

192  * 26  +  8    =  5000

For convenience sake...let the side of the square  = 1

Then the  height of the equilateral triangle  = √3/2

Position D  at  (0,0)

Then A will have the coordinates  ( 1/2, 1 + √3/2)

And we can find  1/2 of the measure of   angle  DAE  as

arctan [  (1/2) / ( 1 +√3/2 ) ]  =  15°

So...DAE  will be twice  this  = 30°

Yep...my bad...see edit....

The altitude of the triangle = 8 * √3/2 = 4√3  = √48

The triangle on the right is a 45-45-90 right triangle

Both  legs  = the altitude

The hypotenuse, x  =  √  [ (√48)^2 + (√48)^2 ]   = √ [48 + 48 ] = √96  = √ [16 * 6 ] = 4√6

There are 90  2 digit  numbers.....the total possible pairs   = C(90,2)

An odd can only occur if we multiply  any two odds together.....the number of possible pairs  = C(45,2)

So....the probability  of an even product  is......  

1  - probability of an odd product  =

1  - C(45,2)/C(90,2)  =

67/89 ≈ 75.3%

EDIT to correct a silly error pointed out by Guest...!!!!