Mark: (you are on fire, that holiday sure put gas in your tank.)
CHESS BOARD SQUARES
204
Number of squares on a chess-board (8 x 8). Then there are 8*8 + 7*7 + 6*6 + 5*5 + 4*4 + 3*3 + 2*2 + 1*1 = 204 squares.
In the same way we can now solve the number of squares on a general n x n board. A similar calculation gives that the number of squares is in that case equal to: n * n + (n-1) * (n-1) + (n-2) * (n-2) + ... + 2 * 2 + 1 * 1 = 1/3 * n^3 + 1/2 * n^2 + 1/6 * n squares.
BLUE & RED DICE
3
If P(Red on 2nd die)=p, then P(Blue on 2nd die)=1−p.
P(same colour)=P(RR)+P(BB)=(5/6)p+(1/6)(1−p)=(4p+1)/6.
But as we are trying to get, P(same colour)=1/2=3/6, it follows that 4p+1=3, 4p=2, and p=1/2; that is, three red and three blue faces.
This can be solved in a surprisingly simple way. Regardless of obtaining red or blue on the first die, the only way we would have equal chance of getting two faces of the same colour is if the second die has an equal number of red and blue faces.
ZIP CODES
Do you consider 2 and 5 to remain the same when rotated as they would on a 7-segment display?
All detour prone ZIP codes will exclusively contain the digits 0, 1, 6, 8, and 9
However, codes like, 66899, are not detour prone, as it reads the same upside down. In fact, any ZIP code with the following criteria will not be detour prone:
iii. 0, 1, or 8, as middle digit.
iv. First/second digit is 0, 1, 6, 8, or 9 (with corresponding last/fourth digit will being 0, 1, 9, 8, and 6, respectively.)
This is a bastich of a problem, and perhaps best avoided.
NOPETS XMAS QUIXZ
Disagreements with Rap's answers"
5) 1" (the small cube was filled twice)
7) 4:q = p:3
8) m/t hours
9) 4 (10 + 2*(-3))
10) 5
(I think Rap put this for (9) by mistake)
"Yes, I'm sure. Remember, this from a guy who said they were too easy"
ART GALLERY
200
The answer is 200. The explanation is that the painting number equates to the number of degrees between a clock's hour-hand and minute-hand (measured in a clockwise direction). The first three examples are easy if you sketch the clock hands on a clock face and plot the hours around the clock face (bear in mind there are 360 degrees around a circle; the 12 on the clock-face equates to zero degrees, and each hour equates to 30 degrees, being one-twelfth of 360).
The puzzle question (9.20pm) is more difficult to calculate than the first three time examples. Here's my attempt to explain it: At 9:20 the minute hand is at the number 4 which is 120 degrees from zero (4 is a third of 12, hence a third of 360 degrees is 120 degrees). The hour hand is at a position equating to 560/720 minutes (there being 720 minutes in 12 hours, and 9hrs 20mins being 560 minutes). 560/720 equates to 280/360 (360 is half of 720, and half of 560 is 280), so the hour hand is at 280 degrees from zero (remember zero is 12 on the clock face). Measured in a clockwise direction, the number of degrees (or angle) between the hour hand and the minute hand is 80 degrees around to the 12 (at zero degrees), plus 120 degrees from the 12 to the 4 (which we previously established). Then simply add: 80 + 120 = 200.
DRIVE TO GUS
infinitely fast
To average 30 miles an hour over two miles, the car must complete the two miles in 4 minutes....
If the car travels at 15 miles an hour for 1 mile, it has take 4 minutes already!
2 miles at 30mph takes 2/30 hours, ie 1/15 hours, ie 4 minutes. Since he has already been driving for 4 minutes there is no need to do any more calculating... it's impossible for him to do it.
RALPH
13 5/7 miles
I have 13.7 but what the heck, it's Christmas.
The Eiffel tower in Paris, including the TV antenna, has a height of 321 meters. It is made out of 7 million kg steel. Francis would like to build a scale model of this tower with a weight of 1 kg.
How tall will this scale model be
