bonanova

[1] The volume of a sphere can be reduced only by reducing its radius, as it remains a sphere. In the puzzle, the drilled sphere is no longer a sphere - it's like a bead that can be put on a string. The drill removes some volume of material from the sphere. The question asks what is the volume of the remaining portion of the sphere. [2] Mass removed is proportional to the volume removed, so density remains the same. Density has nothing to do with the puzzle, which is purely geometric. [3] Surface area does not necessarily increase. It may actually decrease if the sphere is originally large enough.

It certainly seems so at first glance. But are you certain?

Bingo!

Nope. [1] It doesn't obey the rules, and [2] it doesn't solve the problem. [1] You may remove only one item from one box. [2] You have to label all three boxes correctly.

In the land of Knights, Knaves and Liars [KKL], Knights can only tell the truth, Knaves alternately lie and tell the truth [TLT or LTL], and Liars can only lie. Except when they are stricken with the Strange Averitas Flu Virus [sAFV] and become sick. Strangely, as the name implies, when stricken with SAFV, natives of every position behave differently: Knights behave as Knaves; Knaves behave as Liars; and Liars behave as Knights. Again, you are sent to KKL with a limited supply of the costly SAFV vaccine serum. [edit for accuracy et.seq.] To be certain none of the precious medicine is wasted, you must determine which of the KKL natives are truly sick, and administer it only to them. You interview a group of 6 KKL'ers, conveniently named A, B, C, D, E and F. They are eager, if somewhat cryptic, in their attempts to advise you. Their eighteen statements, in valid time sequence, follow: A1. Anyone who, regardless of his state of health, can claim "I am sick" must be a Liar. A2. B is not a sick Liar. A3. I cannot claim to be a Liar who can tell the truth. B1. I am not sick; but take that with a grain of salt, because I am also a Liar. B2. D is a healthy Liar. B3. I am a Knight. C1. Everyone, on becoming sick, must change his story about not being a Knight C1. Everyone, on becoming sick, changes with regard to his ability to claim he is not a Knight. [edit for accuracy] C2. Everyone, on becoming sick, must change his story about not being a Liar. C3. Don't worry about me changing my story, though, I already need a shot. D1. If a person who needs a shot claims that he is not a Liar, then he is indeed a Knight. D2. It is only true of those persons who need a shot that the claim "I am not a Knave" may be made by a person of any position. D3. You can deduce that I, in fact, am not a Knave. E1. Anyone who, regardless of his state of health, can claim "I am a Knight who can lie" must be a Knave. E2. Collectively, the six of us are making equal numbers of true and false statements. E3. Only one of us cannot claim not to be a Knave. F1. Anyone who, regardless of his state of health, can claim to be a Knight must be a Knight. F2. I am a Knave, and I really need a shot. F3. There are at least two of us who have the same position and state of health. To whom do you give shots? Identify the Knights and Liars.

Nice puzzle. Why not hold off posting the spoiler for a day or two on these?

They are anagrams of song titles. They all start with the same letter. For example, No Gloves could mean Love Song.

Perhaps all questions are answered on this. But it occurred to me it's easy to show the sum of squared distances to diagonally opposite corners are the same. Let horizontal and vertical lines through the cow divide the fences into lengths u and v vertically and w and x horizontally. Squared distance to one corner is u2 + x2 and to the diagonally opposite corner, v2 + w2. Squared distances to the other two corners are u2 + w2 and v2 + x2. The squared distances in both cases add to u2 + x2 + w2 + v2.

Thanks. I kept reading to this.

Name them ... [1] A Modern [2] Decade Pruner [3] Cancan Deed [4] A Bravely Remedied [5] A Descended Dirtied Trophy

Great thinking ... but you give me more credit than I deserve. The title doesn't have any intentional clues. B85 had it.

There's a way to pair these words into 5 groups, where each group is the anagram of an 8-letter word. There are a lot of combinations, but each one you find narrows your search for the others. Don't look for a logical connection in the pairs, there may not be one. Can you find all 5 words? PEER TIME DATE WALL JUNE TONE SNOB SEAT PALE VILE

Eleven words need your help to reach completion. Can you help? You'll need all 26 letters, and you can use them only once. _e_er _ _eue _ _o ma_ _p_a_e_ _erso_ _ _ _k_am_on _ouse _a_ _ur_ _ _ _igent

What seven-letter words have the following endings? [1] _ _ _ _ igy [2] _ _ _ _ aoh [3] _ _ _ _ erg [4] _ _ _ _ ynx [5] _ _ _ _ lth [6] _ _ _ _ mox [7] _ _ _ _ ovy [8] _ _ _ _ esh [9] _ _ _ _ awe [10] _ _ _ _ ipt

Yeah, I guess. Wondering why, tho.

A while ago a brilliant scientist, a colleague, opined to me that in the face of widespread bad behavior, it would be a good thing if we could implant a microchip in everyone's brain - essentially removing personal choice regarding legal and moral behavior - and in that way make the situation better. I thought for a moment and then asked him who would write the program. Not me, thanks. My plate is pretty full guiding my own actions. It's fun to think and discuss, tho ... interesting thread.

<soapbox on> I believe it's time someone spoke out against the zero bashing that appears all too often in this and other forums. So here goes ... Zero is too often just misunderstood. He is the measure of a thing, not the absence of a thing. For example: When the Yankees beat the Red Sox 4-0, it's not the case that the Red Sox have no score. They just have no class. They have a score; it's zero. Go Yankees. If something is standing still, it's not the case that it has no speed; rather, its speed is zero. In the number 5207, it's not the case that there is no indication of the number of tens in the number; rather that the number of tens is zero. <pause to catch breath> Placeholder? ... Humbug! Zero is more than that. So very much more. Zero is the identity element for addition, just as unity is the identity element for multiplication. He's an integer of good standing; he's a member of the Real Number club; and he occupies a central position in every coordinate system most of us will ever use. So what if it took a few centuries to learn how to deal with him? That places him in the elite company of atoms, quanta, Game Boys, quarks and four dimensions. So, whaddya say, fellas: Let's embrace him. Include him in our games. Invite him to dinner once in a while [he doesn't eat much]. And not be embarrassed to be seen in public with him. Maybe not ask him to be a denominator, tho; he's a little unsure of himself in that role. <soapbox off>