Infinitely many coins coins are aligned, with heads facing up.
1 – Then, all coins are flipped.
2- Then, the second, fourth, sixth, … are flipped.
3- Then, the third, sixth, ninth,… are flipped.
…
k- Then, coins numbered k, 2k, 3k, … are flipped.
…
After this infinite process is done, what is the proportion of the coins numbered p, p^2, p^3, p^4, … that will be facing heads (where p>1 is an integer) ?
Given the string 123456789101112, in how many different ways can you place “+” signs among some of the digits, so that the overall expression sums up to 2013?
The answer is at least 1, due to Zorglub’s example: 12+34+56+789+10+1112 = 2013
Of all integers, what is the percentage of them that contains the number 3?
ei. 1-10 has one three, 1-100 has 19 three’s…
A quadrilateral has sides of length 1,2,3 and 4, not necessarily in that order. How large can its area be ?
Find all positive integers N, s.t., from the “N string” 12345678910111213….N with the help of (unlimited number of) “+” signs we can derive 2013.
Example: We can derive 28 for N=4 via 1+23+4
Here’s one from Slavy.
All the numbers from 1 to 2012 are written in a string,
i.e. 123456789101112…..20112012. You are given as many pluses “+” as you wish, that you are allowed to place anywhere in the string. For a given positive integer n, greater than 1, in how many different ways can you place the + signs, such that the derived arithmetic expression equals 2013^n?
For clarity, with the numbers from 1 to 4, we can get:
1234, 1+234=235, 12+34=46, 123+4=127, 1+2+34=37, …
During a game of dice, a spectator decided to keep track of the rolls. He recorded the product of the rolls. For example: a roll of 4 and 5 is scored as 20.
1st roll score – (we do not know)
2nd roll score – 5 more than the 1st roll score
3rd roll score – 6 less than the 2nd roll score
4th roll score – 11 more than the 3rd roll score
5th roll score – 8 less than the 4th roll score
What numbers were rolled on each of the five rolls? (Sadly, there are two answers.)
posted on behalf of jenny.
a rich guy lives in a round house and goes out one day. when he comes back his son’s been murdered. he goes up to his baker and asks did you kill my son. the baker says no and that he has been baking cupcakes. the man goes up to to his gardener and asks her is she killed his son and the gardener says no i have been planting you plants all day. the man goes up to his maid and asks her did you kill my son, she said no i have been cleaning your corners all day. so the guy goes up to his pool cleaner and says did you kill my son and he says no i have been adding chemicals in your pool.
whodunit?
The earth is spherical with a radius of 6400 km.
A long chain is tighten around it. Then, an extra piece measuring 1 m is added to the chain and a point of the chain is pulled perpendicularly away from the earth until the chain is tight again. What is the distance between the earth and the point ?
In the Antarctic, below frozen ice sheets I saw things I thought were very odd. Beginning at the bottom of the floating ice sheet, traveling to the sea floor were “branches” of ice. These branches had smaller “limbs” that were also pointed downward. They had small pieces that looked like small fern leaves. Being ice, they were very white.
How could these strange ice branches be reacing for the sea floor be formed?
