bonanova

A while ago for a minimal set of coin denominations that would create any value less than a dollar with at most two coins - i.e. change we could believe in. Today we're asking about change from a slightly different perspective, keeping the coin denominations of dollar, half-dollar, quarter, dime, nickel and penny. For our international solvers these have respective values of $1.00, $0.50, $0.25, $0.10, $0.05, and $0.01. Using as many of each of these coins as may be needed, how many different sets of coins are worth exactly one dollar? Examples: 1 dollar coin - one could represent as: [1 0 0 0 0 0]1 half-dollar, 1 quarter, 1 dime, 2 nickels, 5 pennies - [0 1 1 1 2 5]100 pennies - [0 0 0 0 0 100] etc. For those who distinguish combinations and permutations, we're talking about combinations.

For starters, check out a for f(x) = 2.

Edit: Kudos to parik - I took too long to get it.

Wouldn't it be the case that if Delta=0, the counterfeit coins cannot be identified? OP states there are counterfeits; their weight is within limits that include the weight of genuine coins. Although it could be determined that Delta = 0; weighing would seem powerless to identify the counterfeit pile. That is, the counterfeits could be struck from material of density equal to that of the genuine coins. They would still be counterfeit. I wrote a tongue-in-cheek a while back, after feeling it had been abused by several posters, almost to the extent that it was not given proper status as a "genuine" number. OP says only that Delta be any real number between 5 and -5. Zero is a real number; OP says we should deal with it.

Lovely. You have a trivial typo "I will take 2 1 coins from piles 11 and 12." And I'll happily make the change at your request. - bn

For many of us who love puzzles of the type posted in this Forum, Martin Gardner was a hero. He died a week ago at age 95. For 25 years he published the column Mathematical Games in Scientific American and almost single-handedly popularized the field of recreational mathematics in the U.S. He was also the author of more than 70 books on the subject, many of which compile his puzzles by subject and can be found for sale on-line. More than just the puzzles themselves, though, was his amazing ability to clearly pose them. As anyone who has posted in this forum knows, including myself, it's an art to exclude the thousand- and-one silly answers that a loosely worded puzzle can permit. Read here some tributes written by admirers from around the world.

Not a problem. Done it myself even. You can't always find the right search words. Just wanted to bring the past discussions to the table, along with the great drawings from the previous poster. It's a fun problem, and, have we found a consensus?

This classic problem was at in a discussion of a How do the solutions compare?

In a previous puzzle we are told that in a two-child family one of the children is a girl and then asked the probability that both children are girls. The original [older younger] gender distribution has four cases, each with probability of 1/4: GG GBBGBB Knowing only that "one of the children is a girl" removes the fourth case, while leaving unaffected the relative probabilities of the first three. So the desired probability is 1/3. But clearly there were other true statements our reporter might have made. He may have been able to say one child is a boy. He might have described the older child, or the one nearest to him. What if we knew more about how the statement was chosen? Does the "filter" the reporter uses affect the answer? Let's check that out by restating the problem differently: A reporter with access to the family is allowed to select any true statement of the form "One of the children is a X" where X is boy or girl. We then desire the probability p[same] of both children being X. Clearly before the reporter speaks, p[same] = 1/2. Now he says "One of the children is a girl." If that's all we know, p[same] = 1/3, as in the original puzzle. But now we ask the reporter: "How did you select your statement?" He replies: Before I met the family, I decided that I would ... Let the children play Go Fish and then describe the winner. Describe the older child.Say "one of the children is a girl" if I could; otherwise say "one of the children is a boy." For each of these algorithms, what value of p[same] follows from the reporter's statement?

dawh, Yeah, my idea is very close to what you said. But I think there is a specific observation that nails the ambiguity precisely to this case and not to next door, next year, next subject, next room, or next anything else.

Thinking more specifically: It seems strange that "next month" can belong to "this year," "next week" can belong to "this month," but "next Friday" cannot belong to "this week." Can you think of a logical reason why this usage might have come into existence only for days? That is, a reason that would not lead to next week meaning week after next?

Precisely Martingale.

Something like Martingale.

Suppose it's Sunday April 7. Someone says something will happen next Friday. Invariably they feel they are referring to Friday April 19, and not to Friday April 12. Yet the dictionary makes it clear that next Friday is April 12. Dictionary says this of next: immediately following in time or ordernearest in space or position;immediately adjoining without intervening spaceat the time or occasion immediately following In other words, it's the superlative of near: near, nearer, nearest or next. Thus we use it to express the absence of intervening objects or events: Next in line.Next door.Next year.Our next President.In the next room.Next, he addressed the problem of health care. People have no problem using next correctly in any of the listed case above. Only, it seems, when next is applied to a day does it become common to insert an intervening object. Next Friday is taken to mean Friday after next. Amazing! If you recursively apply that definition, you get, in turn, Friday after Friday after next, then Friday after Friday after Friday after next ... and so on. In other words, Next Friday, if it's even defined, will never come! It's like "Tomorrow, Tomorrow, I love you Tomorrow, you're always a day away." Next Friday is always [at least] a week away! Suppose it's Sunday April 7th. If next Friday means April 19, then it's not the Friday that immediately follows in time, as the definition of next says it should. Rather, there is an intervening Friday. What do we call that Friday? Well, most would call it This Friday. What's that all about, and why is "Next Friday" so ambiguous, when next week or next month or next year is not? I have a theory that I'll share, after some cracks have been taken at this. Edit: see post 7.

So I was reading in Wikipedia about expectation values, and they analyzed roulette. There are 38 outcomes and guessing it pays $35. So on average you lose about a nickel on a dollar bet. So it's dumb to play for a long time: the law of large numbers sends you home broke. Which reminds me of the guy in desperate financial trouble who went to Vegas to gamble. On the way, he asked for supernatural intervention: "Lord, help me to at least break even. I need the money." But I digress. I could win on the first spin of the wheel. Or the second. Or the third, and so on. By the 38th spin, I should expect to have won at least once. But don't the odds favor the case where my one win will occur before the 35th spin? If so then there exists a moment in time when I will be ahead. And I win, just by quitting. So doesn't that mean I can win, in the short term at roulette? http://brainden.com/forum/uploads/emoticons/default_wink.png' alt=';)'> And if that works, couldn't I do it again, and win in the long term? An answer should include an idea of the number of spins I should expect to wait before winning once.

Pi does not have an exact finite decimal representation. But if you perform a change of scale, from integers to circumferences per diameters, it has the very tractable and precise value of unity. Exactly filling the volume might remain impossible, tho, as it could become difficult to count paint molecules in these units.