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I am a math professor.

Proof 3: Factors of x^5-x

Proof 1: Modulus 10 Proof 2: Modulus 5 I have a third proof idea... I'll see if I can flush it out later. Are you a math professor/researcher? If not, where do you find / get inspiration for these?

Assume that n is a natural number, prove that n and n5 will always have the same one's digit. e.g. 13 and 135=371,293 both end in 3.

Constant fraction of current money to maximize the median/mode: Code output (assumes infinitely divisible money): A little more math:

Hi,My team is developing a new online Sudoku game, and to make it a fun, challenging, and seamless experience we are collecting responses from avid Sudoku enthusiasts. Kindly spare 2 mins of your time to fill out this survey form:https://docs.google.com/forms/d/e/1FAIpQLScU_LFBpahyp_rXUHg5...Your responses will be highly appreciated.Regards.

Yes, you have the basic procedure which is to start at or near one end and move 3 holes each time until you reach the end and repeat at or near the start. There are subtle modifications as you describe but they all give about the same result. The interesting point is that there is an enormous difference between this approach and offhand initial guesses.

It seems my initial guess and idea about how the probabilities would end up were... umm, bad. Yay code... My best approach so far... It seems we calculate the expected value for the day the rat is found differently.

Yes to all your questions. There are better ways than your initial guess.

Is this just spam? The link at the bottom of that page seems shady.... installyourfiles dot com? No thanks. It looks another like this was posted recently too. "can anyone out there solve this puzzle?" -- that is also hosted on sites.google.com and goes to getafilenow dot com.

Hey Bob, welcome to the den. It's been a while since I've checked this site. This puzzle, at first glance, seemed similar to "Groundhog in a Hole." -- http://brainden.com/forum/topic/11943--/ I have posted a couple variations, too. -- http://brainden.com/forum/topic/12010--/ and http://brainden.com/forum/topic/18524-groundhog-in-a-hole-yet-again/ Anyway, I thought I'd ask for some clarification. If the rat is in hole 10 on day 1, does that mean on day 2 it has an equal chance (20% each) of having gone to holes 8, 9, 10 (stayed there), 11, or 12? If the rat is in hole 2 on day 1, on day 2 is there an equal chance (25% each) of being in holes 1, 2, 3, or 4? And a third chance each for holes 1, 2, and 3 if starting right on the edge in hole 1? And does the rat have an equal chance of being in any given hole on day 1 (1% each)? Just thought I'd nail down the way the probabilities move. Here's a quick guess before checking... anything really...

Can you solve this puzzle

As this is an open challenge question I want to give other people plenty of time to respond.

I give up. What is the answer?

Medusa

xx

Looking in a middle hole every date has an expected time of over 1,000 days to find the rat. There's a way to do it in less time.

A rat is in one of 100 holes that are in a line. Each night he moves to the left or right 0,1, or 2 holes randomly selected. You pick one hole each day to look in until you find him. What procedure do you use in order to minimize the expected number of days before you find him?

I'm using the generally accepted usage of logical terms: 1. "or" means and/or 2. "if" means "if" ---- I do not mean "if and only if" 3. Given: if A then B. If A is false then nothing can be determined about B. However the contrapositive condition should also be considered: if not B then not A.

Can I ask for a bit of clarification? There are a few things that could be interpreted in a couple of different ways When you say "or", do you mean AND/OR or Exclusive OR? When you say "IF" do you mean simply "IF", or "IF and only IF"? If the IF condition in a character's statement is not met (e.g. A says "If B is lying, then C is involved", but B is actually telling the truth), is that character considered to be telling the truth, lying, or neither? (obviously this becomes clear if the IF statements are interpreted as 'if and only if', but I'm asking in case they are not)

"a" =3 3*3=9, and 9+26=35 35 is NOT a prime number. In conclusion, 3 is a possible answer.

You are the detective in a crime. From your research you know that Jill and John are lying however the others may be telling the truth or lying. What can you deduce from the statements below? Jill said: IF John is not involved and Joe told the truth THEN Mike is not involved John said: IF Abby is involved or Sue is involved THEN Joe is involved Abby said: IF Cindy is not involved or Joe is not involved THEN Mike is involved or Jill is not involved Cindy said: IF Mike is involved and Joe told the truth THEN John is not involved or Jill is involved Sue said: IF Joe is involved or Sue is not involved THEN Cindy is involved or Abby is not involved Mike said: IF Cindy is not involved and Abby is involved THEN John is not involved or Jill is not involved Joe said: IF Jill is telling the truth or Cindy is lying THEN Sue is telling the truth or Mike is telling the truth

I looked at just a few basic strategies. Best I could do was to bet .57 x money-in-hand each bet. What's your best?