bonanova

In partition theory, a counting number n is expressed as the sum of positive integers, without regard to order. If n is a triangular number it has a special partition, namely 1+2+3+4+ ... Examples of triangular numbers are 1, 3, 6, 10, 15, 21, 28, 36, 45, 55, and so on. If we select a triangular number of cards from a deck of 52, the largest number we'd have is 49 45. Let's go with 49 45cards [any 49 45cards, it doesn't matter] and play a little game. Deal the cards into some number of piles, each pile having some number of cards. Doesn't matter how many piles - could be a single pile, could be 49 45 piles. Your choice. What you've done is create a partition of the number 49 45. Play proceeds as follows: . Remove 1 card from each pile.Place these cards together and use them to form a new pile.. For example, if you began with 49 45 piles of one card each, after the first play you'd have a single pile of 49 45 cards. After two plays you'd have a pile of 48 cards and a pile of 1 card. And so on. Piles are created and sometimes piles are used up. Continue repeating steps 1 and 2. The puzzle is to find the answer to these questions: Does the game ever get stuck in an endless loop? i.e. Does a partition ever revert to a previous partition? .Does the game ever reach a stable state? i.e. Is a partition ever reached that is invariant to steps 1 and 2? .In answering these questions remember that, in a partition, the order does not matter. So in a game of 3 cards, piles of 1 1 and 1 card and a single pile of 3 cards are different partitions. But piles of 1 and 2 cards and piles of 2 and 1 cards are the same partition. Shuffle up, deal and play the game, or just think about it. Either way, enjoy!

Each of three circles passes through the centers of the other two. The area interior to all three is quite close to, but different from, one quarter of the area A of a circle. Can you determine, through inference, construction or calculation, whether it's smaller or larger than A/4? Caveat: Arrgh. Forgot that I posted this problem before;* where it was solved by calculation. I'll bend the Den's rules a bit and leave it up, but modified. Please provide a purely geometrical solution. *A benefit of advancing age is the ability to hide one's own Easter eggs.

Hint: Use code tags to align tables and grids (any font is ok): (code) a b c d e f g h i j k l m n o p q r s t u v w x y (/code) But use [] brackets instead of parentheses. a b c d e f g h i j k l m n o p q r s t u v w x y [/code]

It may have improved the puzzle to add: Determine the truth values of these four statements. Then give [with statement D as an example] an appropriate response to each. I don't mean to prolong a discussion about form, just guidance to help solvers get at what the puzzle is asking. SP: Understand. Intractable was meant to point to OP not to a deficiency in the answer. The candidate answer pointed to uncertainty of the OP. And if truth values for A-D were the object to be determined, that does open the can of peas as to truth or falsity on what level, when the statements are compound. Usually the highest level.

Hint: Answer spoilers are optional. You might want to more strongly isolate solvers from the solution, or just give a hint.

If the speaker were a sphere, sure ... good one.

If what you want is answers, it's fairest to the problem solvers to ask questions. A good puzzle defines the requested response. One thing you might do is change the instructions to statements [conditionals] and ask for a consistent set of truth values. If that's in fact what you desire, why not explicitly ask for it? Otherwise you're left with something like this: If 2+2=5 then New York is a small city. If I were to call that a question, what answer would you give? Imperatives do not have truth values: Go sit in that chair. True? False?

I agree with your point [npi] that a line segment includes its interior; a degenerate triangle therefore does not comprise a 1-dimensional case of three objects that touch only at three distinct points. But does a circle consistently collapse to a line segment? Consider: A sphere is a surface - the locus of points in 3-space equidistant from a center point. A circle is a curved line - the locus of points in 2-space equidistant from a center point. It is consistent, then, for a circle to collapse to the locus of points in 1-space equidistant from a center point. That is, a circle collapses to a point pair. Taking this view, collinear point pairs AB and BC, even when point B is the leftmost or rightmost of the three, touch only at point B, and can be mutually intimate at three distinct points with point pair AC. The interior volume of a sphere [which does not belong to the sphere, rather to a ball] collapses to the interior area of a circle [which does not belong to the circle, rather to a disk] which collapses to the interior length of a point pair [which does not belong to the point pair, rather to a line segment. A sphere's exterior, a 2-dimensional [in spherical coordinates] surface area, becomes a circle's exterior, a 1-dimensional [in radial coordinates] perimeter line, which then becomes a point pair's exterior, the 0-dimensional [in any coordinates] points.

Simulation - the lazy person's way of calculating probability - shows that

Can anyone imagine an intimate configuration for n=1? i.e. 1-dimensional space.

Nice approach

That's it. Better, even, than the stock answer of equal mod 194.

I'm betting on that answer. Nice!

Very cool, Bonanova. I like. Glad you liked it. If you like your sig, maybe you'll like

Not sure about araver, nicely done on [1] and [3]. I'm wondering whether you're saying that Thanksgiving, because of leap year, does not occur on the 27th every seven years on average? On different years it's Nov 22 thru Nov 28. In a given year is one date more likely? Hint: [2] is not a "base" relationship; it's a rather ad hoc equivalence included to make a set of three.

Well, one could say vanishingly small and remove this objection. For a short, tongue in cheek treatise on the existence of things with attributes of zero,

I guess the answer to that would be a sphere with a hole is not a sphere.

If Roses are red / Violets are blue can scramble into RARVABOREIRLSEDOEUELESETS, what lines of poetry would similarly become TINFLABTTULAHSORIOOASAWEIKOKNARGEKEDYEASTE?