# CTK Insights

• ## Pages

31 Dec

### Angle Bisectors In Rectangle

TweetABCD is a rectangle; M and N are the midpoints of sides AD and BC, respectively. Let P lie on CD, Q be the intersection of MP and AC. Prove that MN is the bisector of angle PNQ

27 Dec

### Palindromic moves in the Toads and Frogs Puzzle

TweetIn the Toads and Frogs puzzle the sequence of moves in the Toads and Frogs puzzle is always palindromic. To undertsand why this is so needs very little in case the number of toads is the same as the number of frogs. But this is also true even if the two numbers are different. Why?

21 Dec

### Is 0.999... = 1?

TweetAnother look at the perennial stumbling block of whether 0.999... = 1 and what it actually may mean, see http://www.cut-the-knot.org/WhatIs/Infinity/9999.shtml

20 Dec

### Quotes

TweetLeisure is the mother of philosophy. And further, From this it was that the place where any of them (AB: philosophers) taught and disputed was called schola. which in their tongue signifieth leisure; and their disputations, diatribae, that is to say, passing of the time. Thomas Hobbes Leviathan, ch. 46 Penguin Classics, 1982 Gentelfolks in […]

14 Dec

### Circles, Products and Optimization

TweetHere's a lively problem that was offered at the 1975 USA Math Olympiad (Porblem 4): Circle C(E), with center E, and C(F) with center F, meet in points P and Q. A on C(E) and B on C(F) are such that AB passes through P. Find the position of A and B for which AP×PB […]

13 Dec

### There is nothing new under the sun

TweetI am still reading A. Einstein's collection Ideas and Opinions. Some of these seem to be naive but most I agree with. Here is one written some time in 1934: The power of conscience and of international spirit has proved itself inadequate. At present it is being so weak as to tolerate parleying with the […]

12 Dec

### The size of the error

TweetI was surprised to come across the following The fact that on the basis of such laws we are able to predict the temporal behavior of phenomena in certain domains with great precision and certainty is deeply embedded in the consciousmess of the modern man, even though he may have grasped very little of the […]

09 Dec

### Rabbits Reproduce; Integers Don't

TweetMathematical induction fallacy "proves" that all rabbits are of the same color. Edward Barbeau quotes a computer science student who offered this refutation: For the basis of induction pick any one rabbit. By default, the rabbit is of the same color with itself. For the inductive step, assume that any set of fewer than n […]