# Men digging holes

Suppose 9 men take 9 hours to dig 9 holes in the ground.

(a) How many hours would it take 3 men to dig 3 holes?

(b) How many men would it take to dig 54 holes in 54 hours?

• This is a simple application of the rule of three (mathforum.org/library/drmath/view/60822.html) and shouldn't even be considered a puzzle. – Guntram Blohm supports Monica Oct 10 '14 at 15:48
• It's a trick puzzle, since people tend to assume that all proportions apply simultaneously. – Joe Z. Oct 10 '14 at 17:49
• Super simple puzzle, but its still fun. – DiscOH Oct 10 '14 at 18:04
• How many hours would it take 5 men to dig half a hole? – James Oct 10 '14 at 23:53

a) 9
b) 9

9 men take 9 hours to dig 9 holes =>
1 man takes 9 hours to dig 1 hole =>
a) 3 men take 9 hours to dig 3 holes
1 man take 9*54 hours to dig 54 holes =>
b) 9 men take 54 hours to dig 54 holes

• They are quite extraordinary, these men who can work 54 hours straight without rest or sleep (or breaks for food, water, or the bathroom). – David Conrad Oct 10 '14 at 19:10
• @DavidConrad 54 hours of work, not straight. – Scimonster Nov 6 '14 at 15:25

A) 3 hours.

The men are all in a line and by force of habit just throw the dirt behind them. Unfortunately that falls into the hole of the person behind them. So the first one finishes in 1 hour, the second (now that no one is putting dirt in his hole) finishes an hour after that. There's only room for 1 man per hole so they can't go helping their friends, so the last guy ends up digging a hole for each person.

These men really need a manager.

B) 54 men. As stated above, they each dig and dig and dig as the idiot in front of them fills in their hole each time they take a shovel full out. Remember if it was just one guy, either he would make a big pile where the next hole is supposed to go, or he would dig a hole, walk forward and dig his hole out again, and have to redig it! That would be terrible. It would take this guy 1485 hours to do it by himself because we have to assume they send the worst one out there.

• Best Answer IMO – DiscOH Oct 10 '14 at 18:03
• I don't think "54 hours" can possibly be the answer to the question "How many men would it take...". – David Conrad Oct 10 '14 at 19:08
• Haha you're correct, I shouldn't have put labels on there necessarily. – corsiKa Oct 10 '14 at 19:12

1 hour (or less).

As soon as the men start digging there is a hole in the ground, so all you need is for each man to start digging their own hole and your done. It may not be as deep as the hole that the 9 men dug when they spent 9 hours digging, but it's still a hole. The puzzle doesn't specify that the new holes have to be as deep as the old ones.

I believe the answer for part a is one hour, as it takes 9 men take 9 hours to dig 9 holes, then the 9 men dig one hole per hour (9 holes/9 hours = 1), and cosequently one man dig 1/9 hole per hour (1 hole / 9 men = 1/9). Now if there are 3 men then they dig 1/3 of a hole per hour (3 * 1/9 = 1/3) and they are going to work for 3 hours then they dig one hole (3 * 1/3 = 1). Part b is correct I hope this make sense

• I think everything you said is precise. But the conclusion is not one hour, but nine hours. – FIreCase Jan 5 at 19:09
• "they are going to work for 3 hours then they dig one hole (3 * 1/3 = 1)." Yes, but the question requires them to dig three holes, not one hole. – ZanyG Jan 5 at 23:00