Yeah, the farthest you can walk into the woods is halfway through the woods. So if someone says “how far can you walk into the woods” and the answer is “halfway”, that means you’re only a quarter of the way through the woods, doesn’t it?
I seem to have created a hijack of my own thread. I’ll add my own two cents and then let it go.
Every complex number has exactly n n-th roots. As you pointed out, the number 9 has two square roots: 3 and -3. The square root function y=sqrt{x} is, however, not a multivalued function. It is defined, for positive x, as the positive solution of the equation y^2 = x. (Or, if you like, the square root function y = sqrt{x} is the inverse of the function y=x^2, x >=0). There are two solutions to the equation y^2 = 9. One of them is the square root of 9. The other one is -3.
3 and -3 are square roots of 9.
3 is the square root of 9. The symbol sqrt{9} means the number 3.
The square root of x^2 is |x|.
This is analogous to the definition of the inverse sine function: for any value of y in between -1 and 1 there are infinitely many solutions to the equation y=sin(x). The inverse sine of y is the unique solution of this equation in the interval [-pi/2,pi/2].
Might possibly be too hard for a 10-year-old, but…
Two women apply for the same job on the same day. They look exactly alike. The boss looks at their applications and sees that they have the same last name, the same address, and the same birthdate. “Wow,” says the boss. “You must be twins.”
A man builds a house with a house that has four sides, all facing south. A bear walks by. What color is the bear?
White. The house has to be at the North Pole
Two men are playing checkers. They’ve played five games and each man has won the same number of games (NB – there are no ties in checkers). How is this possible?
They’re not playing each other.
What word is always misspelled in the dictionary?
Misspelled
Divide 20 by one half and add ten. What’s the answer?
The punch line for that was on The Office, I always wondered what the joke was.
Regarding the boat: I got the answer, then I thought too hard about it and wondered if the anchor would keep the boat at the same height relative to the sea bed.
Not necessarily. The anchor will prevent the boat from rising with the tide if the anchor weighs more than a volume of water equal to the volume of a horizontal cross-section of the boat extending from the waterline to three feet above the waterline. An extremely implausible scenario, but possible nonetheless if you have a nutcase designing the anchor. However, that does not guarantee that the anchor weighs more than a volume of water equal to the volume of the boat’s hull.
In other words, as you try to raise the anchor you’ll pull the boat lower in the water. But as it gets lower it will displace more water and become more buoyant, and will eventually be able to pull the anchor off the seabed.
True, but in practical terms it’s not the anchor that holds you in place, it’s the chain. In 30 feet of water you might drop the anchor and 150 feet of chain. The anchor lays on its side and hopefully digs in a little bit; 120 feet of chain spreads out from that, and it’s only the last 30 feet that’s weighing down on the ship. If that 30 feet of chain is enough to keep the ship from rising with the tide, you’re going to need to gain a lot of buoyancy to haul up 150 feet of chain, plus the anchor.
OK, scratch that. The ocean floor at low tide is only three feet below the lowest point on the boat’s hull, and there’s a hole in the hull one inch above the waterline. You gotta be ready for anything a ten-year-old can shoot at you.