## Digging in the Dirt

Wow, this one is tricky-see if you can figure it out:

A boy comes upon a man digging a pit.

“Good morning,” he said. “How deep is that pit?”

“Try to Guess”, replied the man. “My height is exactly five feet ten inches.”

“How much deeper are you going?” said the boy.

“I’m going twice as deep,” was the answer, “and then my head will be twice as far below ground as it now above ground.”

*Any thoughts? Feel free to submit your answer in the comment section below; will unmask before the week is over.*

michaelc| Profile July 14th, 2009 - 10:58 pmI take when he says “my head”, he means the top of his head.

If we let the the distance above the ground be x, and the depth of the hole equal y, we can see that when he digs it twice as deep, the hole must then equal 2y. And we also see that the top of his head is below the ground by a distance of 2x. Also before he digs we see, 70 = x + y . (the depth of the hole plus the portion above ground is his height)

So the entire depth of the hole is 2y. The man’s height is 70 inches, and the distance from the top of his head to the top of the hole (the ground level) is 2x. So 2y = 70 + 2x.

So a little substitution, and you see that x = 17.5 and y=52.5.

So the hole is 52.5 inches, his head is sticking out 17.5 inches. When he digs twice the depth, the hole will be 105 inches, so 35 inches will be from ground level to the top of the man’s head.

I always found it easier to guess with some math help. Silly me.

Falwan| Profile July 14th, 2009 - 11:39 pmIs that you ,Dr. RK, digging for puzzles?

Falwan| Profile July 14th, 2009 - 11:55 pmpit is 3 feet 6 inches…

depth = x

head above by = 70 – x

head below by = 2(70 – x )

equation to solve :

70 + 2(70 – x ) = 3x

x = 42 inches = 3ft. 6 in.

poopsicle| Profile July 15th, 2009 - 3:58 am70-x=a

2a+70=2x

x=current depth

a=current height above ground

x=52.5in

a=17.5in

bigbossSNK| Profile July 15th, 2009 - 5:01 amVisualizing the man in the pit simplifies the problem.

The man’s head is above ground. His height is 5’10”. Standing in the pit, his height corresponds to the depth of the pit after adding the length his head is above ground.

In math lingo, let x be the depth of the pit, h be the length his head is above ground. Then:

x+h= 5′ 10”

He also says that twice the current depth of the pit corresponds to his height after adding twice the current length between the ground and the top of his head. Again:

2x= 5′ 10” +2h

Elementary, x=4′ 4,5”

(I’m accustomed to the metric system, so if my math or notations are incorrect, point it out)

mrman| Profile July 15th, 2009 - 9:24 am52.5 inches is what I came up with.

NitzOO| Profile July 15th, 2009 - 9:34 amY= man’s height = 5ft. 10 in. = 70 in.

X= pit’s depth

Z= head’s distance above/below ground

The initial circumstance, when the boy comes and questions the man:

Y = X+Z => X = Y-Z (a)

The final circumstance, after the man has dug the pit twice as deep:

2X = Y+2Z => X = (Y+2Z)/2 (b)

From (a) and (b) => Y = 4Z

Z = 17.5 in. = 1ft. 5.5 in.

X = 3Z = 52.5 in. = 4 ft. 4.5 in.

So the pit’s depth is 4 feet and 4.5 inches.

P.S.: Maybe the final result is incorrect; I’m not good with imperial units.

Shawn| PUZZLE GRANDMASTER | Profile July 15th, 2009 - 9:45 amThe pit is 52.5″ deep.

TheMadEgyptian| Profile July 15th, 2009 - 10:02 amThe pit is 52.5 inches deep (4’4.5″, leaving his head 17.5 inches above ground. The final depth will be 105 inches (8’9″ with his head 35 inches below.

Danthok| Profile July 15th, 2009 - 10:07 am52.5″ deep

DanK| Profile July 15th, 2009 - 10:30 amThe man is 5ft 10in or 70in tall and no matter how far down he digs the top of his head will always be 70in from the bottom of the hole. So his goal is to dig until the top of his head is 140in (twice his height) from the rim of the hole or until the hole is 210in (17 ft 6in) which is 3 times his height. If he needs to go twice as deep to reach that goal then he is currently at 105in (8ft 9in).

joe| Profile July 15th, 2009 - 10:38 amWe assume the man is already in the pit (as in the photo and we let the present depth of the pit be x, and the height that he sticks out of the pit be y. This means that his height is equal to x+y.

(1) … x+y = 70 inches

(we will convert back to feet in the end this is easier to work with)

He says he will go down twice the present depth ( 2x) and that his head will then be twice the distance below as it is above (so 2y )

In other words his height plus this 2y will equal the pit height, 2x

(2) … 70″ + 2y = 2x

Solving simultaneous equations (1) and (2) gives

x = 52.5 inches … this is the answer as the boy asks how deep is the pit when he meets him (in feet it is four feet and four and half inches)

Ari| Profile July 15th, 2009 - 12:40 pmA=distance from his head to ground level

B=how much the rest of his torso is under ground (eg. the depth of the pit)

D=total depth when he is finished digging

The dude is 5ft10in in height which equals 70in.

…so:

A+B=70 => A=70-B

D=2A+70 = 2(70-B)+70)=2B

=> 4B=210

B=52,5in.

So when asked, his pit was 4ft. 4½ in. deep and he is aiming for

a 8ft. 9in. deep hole for whatever reason.

Cyberjar88| Profile July 15th, 2009 - 1:02 pmAssuming the man had not yet started digging, the pit would be 11’8″ deep upon completion.

Blusummers13| Profile July 15th, 2009 - 2:04 pmThe current depth of the hole is 52.5 inches or 4 feet 4.5 inches.

The final depth will be 8 feet 9 inches.

Alchemist| Profile July 15th, 2009 - 2:33 pmThe hole is currently 52.5″ deep.

pravinkumar1942@gmail.com| Profile July 15th, 2009 - 6:18 pmi guess 5ft 10 inch

because he head is at ground level now….

guys can u help me 2 solve this???

fuzzy| Profile July 15th, 2009 - 7:37 pmMy answer is not pretty, but here goes.

The man’s hole is 52.5 inches, or 4 feet 4.5 inches, deep right now.

He is 70 inches tall. Right now he is 17.5 inches above and 52.5 inches below the ground. When the hole is twice as deep, or 105 inches, his head will be 35 inches below the ground, which is twice 17.5.

Guurzak| Profile July 15th, 2009 - 8:00 pmx = current depth of pit

70-x = current height of head above ground

2x-70 = later depth of head below ground

2x-70 = 2(70-x)

2x-70 = 140-2x

4x = 210

x = 52.5

test: 70 – 52.5 = 17.5

105 – 70 = 35 = 2 * 17.5 ?

Jimmy Anders| PUZZLE MASTER | Profile July 15th, 2009 - 9:48 pm4 feet, 4 and 1/2 inches, and boy is my skin crawling just thinking about touching all that sand… Eww!

brianu| Profile July 15th, 2009 - 10:08 pmThe hole is 4’4.5″ and will be 8’9″

doubletime| Profile July 16th, 2009 - 12:38 amhow far was his head above ground to begin with? was he in the pit when the boy walked up?

jmart574| Profile July 16th, 2009 - 12:45 amThe hole is 52.5 inches deep.

The man is 70 inches tall and his head is A inches above ground

Hole one is 70-A, hole two is 70+2A

Finding a hole twice as deep will be the equation (70-A)2=70+2A

A=17.5, 70 inches tall minus 17.5 inches above the ground will give you 52.5

TonyTKL| Profile July 16th, 2009 - 7:04 amAssuming he has only started digging, his head will be 10 feet 20 inches below the sand.

RK| Founder | Profile July 16th, 2009 - 1:49 pmWhile I badly need a vacation Falwan, I’m unfortunately nowhere near a beach right now!

RK| Founder | Profile July 16th, 2009 - 10:45 pmmost of our math champs seem to concur the answer is 52.5….

Annastique| Profile July 25th, 2009 - 12:35 amI see all this math so this may be a stupid answer but he says i’m going twice as deep as my head is *above* around and is the hole is already as deep as him (to the top of his head) hes not going any deaper cuz his head is not above ground it is already below.

Annastique| Profile July 25th, 2009 - 12:37 amI see all this math so this may be a stupid answer but he says i’m going twice as deep as my head is *above* ground and if the hole is already as deep as him (or in other words to the top of his head) hes not going any deeper cuz his head is not above ground it is already below.

barryt| Profile July 25th, 2009 - 8:37 amI am 72 years old but I can still remember how to solve simultaneous equations

from 60 years ago!

If the hole is y” deep at the start and the man’s head is x” above the top,

x + y = 70″ (equation a).

When the hole is 3y” deep the man’s head is 2x” below the top, so

3y – 2x = 70 (equation b).

3y + 3x = 210 (equation 1 x 3

Solving simultaneous equations,

5x = 140, hence x = 28, y = 42

So the hole is 42″ deep at present and will be 126″ deep eventually.

RK| Founder | Profile July 25th, 2009 - 1:17 pmVery Nice Work, barryt! Welcome aboard

ben_stilwell| Profile July 30th, 2009 - 2:29 pmD is depth, A the length of man above ground

D = 70 – A

2D = 70 + 2A

2(70 – A) = 70 + 2A

140 = 70 + 4A

4A = 70

A = 17.5

So his head is 17.5 inches above ground, leaving the depth 52.5 inches.

Which works out. When the hole is twice the depth, 105 inches, the top of his head will be 35 inches below ground.

Depth: 52.5

hex| PUZZLE MASTER | Profile September 25th, 2009 - 12:50 pmI don’t know how i missed this one:

x+y=5’10″

2y-2x=5’10″

solve for y=52.5″ (hole depth)