bonanova

I read your post way too fast, and missed [d]. Fabulous...! Now, who's gonna tell the producers?

Really, we are trying to define ALL numbers thus: Now we only found the SMALLEST of the numbers defined by #2, but there are an infinite amount that go above that number, many of which I am sure cannot be described using fewer than 23 syllables when written out in any way. You say that you saw the paradox, but you say we did find the smallest number .... etc. The paradox says that we did not find ... [nor can anyone find - nor is there] the smallest number not describable using fewer than twenty-three syllables. Since that set lacks a smallest member it is the empty set.

If you choose 4, you can ignore that as well. How about "Some of the customers love the food, because you can neither find a specific one that does, nor doesn't." "Some" means "at least one." And there isn't one. Read existential import. You can talk about all of nothing and about none of nothing. But you can't talk about some of nothing.

1. You cannot ask a "puzzle question" to be "precised". 2. Furthermore, if father's age is 46 years old and son's age is 20 years old, the father's age is a multiple of the son's age when we "precisely" count the age into minutes or seconds. Therefore, besides the father is older enough to give a birth, the son could be 10, 12, 16, 20 or other normal ages. Nope, nope, and ... nope. Not unless you assume the question asks the ages in those units for which the multiples are integral -- minutes, seconds. You obviously did not assume that -- years are the only units mentioned in your answer. You're dancing around the issue pretty well, but you're running out of places to hide. Precision or looseness aside, it's nice to be consistent.

For the cut-pieces of a rectangle to form triangles, the cuts must pass thru corners [not sides]. Only two such cuts exist - along the two diagonals. That gives you 4 triangles, not 5. The only other way to cut a rectangle is to cut it after it has been folded. All my tries at this so far have led to an even number of triangles. I'll keep trying.

OP states .... Furthermore, the father's age is a multiple of the son's age. If multiple does not mean integral multiple, the statement gives no information. Every pair of ages, otherwise, could be described as multiples. So if you reject that as precise, let's affirm it for all practical purposes.

How do you determine an order? Without that, what does Nth mean?

Do the cuts have to be straight lines?

OK. It looks like what you do is [1] remove a ball at random [2] re-arrange the other balls into their original order [e.g. ACD in line 3] but ... not ACB in the last line. I don't get that part. [3] replace the removed ball at the beginning [left end] of the line. OK now what are you asking? Can you rephrase it or give an example? Thanks.

Loosely, perhaps, but not precisely: [1] The problem said multiple. That usually means k is not only a constant, but also an integer. [2] That would be step-father and step-son.

Use the ravel operator, then the sort operator, then the shape operator. Ravel linearizes a higher-order array. A linear array can be sorted. The sorted linear array can then be reshaped to that of the original array. At IBM, back in the 1960's, Ken Iverson invented APL "read A Programming Language" that provided operators like ravel and reshape. Google it for more information. IBM still sells a version of APL that runs on a PC.

I claim it's how we describe categories of things. No math needed... [following added in edit] In the case of the restaurant that has no customers, ALL of the customers love the food. Because you can't find ONE that doesn't. and ... NONE of the customers love the food. Because you can't find ONE that does. It's logically ok to use universal quantifiers [all, no, none] with empty sets. But you can't use particular quantifiers [one, some] with empty sets.

OK, the reasoning goes like this. According to common sense and also something called the Well-ordered Principle, any set of numbers can be ordered, least to greatest; the only exception being the empty set. Every non-empty set of numbers has a member which is the smallest member of that set. Next... Using up to N [N is finite] syllables, in various combinations / permutations, only a finite number of numbers can be described. For example, for N=2 those numbers would be 1, 2, 3, 4, 5, 6, 8, 9, 10, 12 [using 1 syllable] and 7, 13, 14, 15, 16, 18, 19, 20, 30, 40, 50, 60, 80, 90 [using 2 syllables] So those are the numbers that can be described using fewer than 3 syllables. If it could be shown that there are no numbers that cannot be described using fewer than 3 syllables, then these would be all the numbers that exist. A finite number. It would be like proving there are no numbers that require 3 or more syllables. Let's see if that's possible to do. We ask, what is the smallest number that cannot be described using fewer than 3 syllables? Well, there is an answer to that. It's 11 - e-lev-en - 3 syllables. There are others, of course, like 17, 21, 22, ... but 11 is the smallest one. So there are numbers beyond those describable using fewer than 3 syllables. But now we ask, what is the smallest number that cannot be described using fewer than 23 syllables. Well, there seems to be an answer to that as well. It's 1,777,777. -- 23 syllables, and no one found a smaller one. Enter the paradox. 1,777,777 was determined to be the number that is described by the phrase the smallest number that cannot be described using fewer than 23 syllables. But that phrase has 22 syllables. Ooops! By that logic 1,777,777 cannot be - nor can any other number be - the smallest number not specifiable using fewer than 23 syllables. That is, the set of numbers described by that phrase has no smallest member. By the well-ordered principle, therefore, that set of numbers is empty. Now let's talk about the set of all numbers. It comprises two subsets: [1] the set of all numbers that can be described using fewer than 23 syllables - a finite set [since 23 is finite.] [2] the set of all numbers that cannot be described using fewer than 23 syllables - by WOP, the empty set. Thus the set of all numbers is finite. What have the great brains of our time done about things like this? They note that the heart of the paradox is that it references itself. Rather, the answer is described on one level, and disallowed by a description on another level. That type of paradox is called self-referential. They deal with it by assigning its statements a level, according to a hierarchy. Then they allow a statement to reference only those objects on its own level of hierarchy. In this case the number of syllables in speaking the number would be on a different hierarchical level from the number of syllables in the phrase that describes the number. That phrase would then not be permitted to disallow the answer found by counting syllables. Bertrand Russel once said, The point of philosophy is to start with something so simple as not to seem worth stating, and to end with somethiong so paradoxical that no one will believe it.

Bravo cpotting and brhan. For totally different reasons.

Barely noticed, the bell rings; and the opponents sit quietly in their corners while their seconds attend quickly to the cuts and bruises. The bell rings for Round 4 ....

Cute puzzle. I think I have the idea. Here's a start, more to come after I get some zzzzzzzzzzz's.

I did this, once, back in school. It was a computer game, and there were colors instead of numbers, and there could be repeats. I'll try to work this out later, but right now my intuition is saying it can be done in ...

Seven to go, and my brain is mush tonight ... 14 15 P in a R T 15 3 W on a T 17 11 P in a F (S) T 24 13 L in a B D 26 9 L of a C 30 9 P in S A 31 6 B to an O in C I wondered whether 17 is 11 Players in [on?] a Football Team ... but what is the S? Not knowing anything about soccer, does that apply?

If your choice is [3], it's correct; but ... Why is it the best answer? Why did you pick it over [1] All even prime numbers [excluding 2] are divisible by 5? Aren't they all divisible by 5? Show me one that is not. [red text edited]

One-mill-ion sev-en hund-red sev-en-ty sev-en thou-sand sev-en hund-red sev-en-ty sev-en = 23. Bravo, Writersblock. 1,777,777 is the smallest number not specifiable using fewer than twenty-three syllables. At least, no one has come up with a smaller number. So let's say it is. You get the prize. O wait. This is supposed to be a paradox. ummm, just for the heck of it, count the syllables in red, above. If the red words specified your answer, then .... Oh ... we never got to the paradox. Let's try again: ummm, just for the heck of it, count the syllables in red, above. the smal-lest num-ber not spe-ci-fi-a-ble us-ing few-er than twen-ty-three syl-la-bles. If the red words specified the answer, then .... The smallest number not specifiable using fewer than twenty-three syllables has just been specified using fewer than twenty-three syllables. Which leads to the conclusion that there is only a finite number of natural numbers. Good old Berry ...

Aha!!! 73. The Dead Kennedys.... [url:d0eb1]http://www.deadkennedys.com/ How sick, by the way ....

hah! yes ... See How They Run! cute.

My picks:

Hmmm.... Ok; so there is a finite number of numbers. No biggie, I guess. After all, there is also only a finite number of textbooks to be re-written.

I can add ... 3 7 Wonders of the World 5 66 Books of the Bible 9 39 Books of the Old Testament 10 5 Toes on a Foot [?] 11 90 Degrees in a Right Angle 13 32 is the Temperature in Degrees Farenheit at which Water Freezes 16 100 Cents in a Dollar 18 12 Months in a Year 21 29 Days in February in a Leap Year 22 27 Books in the New Testament 32 1000 Years in a Millenium That leaves .... 7 13 S in the U S F 12 3 B M (S H T R) 14 15 P in a R T 15 3 W on a T 17 11 P in a F (S) T 19 13 is U F S 24 13 L in a B D 26 9 L of a C 28 23 P of C in the H B 29 64 S on a C B 30 9 P in S A 31 6 B to an O in C 33 15 M on a D M C