Two questions for you:

The first is more mundane, and is about compass and straight edge construction:

A. Quickest, cleanest way to construct a regular octagon.

B. Given a side, quickest, cleanest way to raise a regular octagon on it.

The second:

Where would you cut a unit square to create a regular octagon? (how far from the corner is your first cut?)

We have six statements about a number, and we know that exactly 1 is false.

1. I am greater than 50
2. I am a multiple of 7
3. I am a perfect square
4. I am a 3-digit number
5. I am less than 500
6. I am a multiple of 17

Are there any numbers that fit? How many? And if they exist, what are they?

Unrelated, interesting note: Nice factoring techniques for solving problems such as

Find n such that n(n+16) is a perfect square

I have been participating, on and off, in Monday Math Madness. It is a contest alternating between Wild About Math (Sol) and Blinkdagger. So they are up to #11, I have submitted correct entries to maybe half of them, but so far, not chosen.

#11 asked about rearranging the digits 123456 to create multiples of 11. How many can you make? OK, nice question. I submit a nice statement of solution. Once again, someone else chosen.

(neat note, below the fold –> )

But, after I solved the little problem, I ran on:

As a consolation, all 720 of them are multiples of 3.
Half of them (360) are even (multiples of 2)
One in 6 (120) are multiples of 5.
Eight in thirty (192) are multiples of 4.
None are multiples of 9.
Fourteen of 120 (84) are multiples of 8.
Multiples of 7? New puzzle, good place to stop.

Consolation prize! Sol ran my little ‘extra’ bit. See, sometimes it pays not to know when to shut up! :)

Yesterday I asked:

Which is more likely, rolling three normal dice, and having one be the sum of the other two? or rolling three normal dice, and having one be the product of the other two?

In addition to your regular work, try to find a (good!) explanation that appeals to intuition.

There was a cube dissection problem the other day, all about conditional probability, inspired by this post on Math Notations. The puzzle here was hard. The one at Math Notations was classic.

Here’s something a bit easier. It challenges you to represent some quantities algebraically (if it’s very very easy for you, you might want to discuss without posting a spoiler, at least right away)

A cube has 8 vertices (corners), 12 edges, and 6 faces (flat sides).

If we cut it down the middle three times, each time parallel to a pair of faces, we get a 2×2x2 cube, and each little cube (cubelet is what I like to call them, and now there are 8 of them) is a corner.

If we cut each edge into thirds, parallel to a pair of faces, we get a 3×3x3 cube, 27 cubelets in all, with 8 corners, 12 edges, 6 cubelets that are on faces but not edges, and one center cubelet that we can’t see from outside. This is like a rubik’s cube (except for the center)

Now, the question: if we go wild slicing, and we get a cube that is b x b x b,

1. how many cubelets will there be?
2. how many corners?
3. how many edges?
4. how many face cubelets?
5. how many center cubelets?

Over at Math Notations, Dave recently posed a variation on a cube dissection problem. Here’s another:

A cube is painted on the outside and then cut into 27 equal cubes by dividing each edge in three with four planes which are parallel to the faces of the original cube.

The cubelets are placed in a bag. One is removed at random and tossed. The face on top is unpainted.

If that same cubelet is tossed again, what is the probability that an unpainted face will appear on top again?

I was thinking of asking for a square with perimeter = area, but that’s either silly or boring.

Give me a square, any square. I’ll bisect a side, then bisect that segment, call the result a unit, and the perimeter is 16 units, the area 16 square units. Works for any square. Boring.

So, readers, can we do something interesting with this? Different shapes? Different regular polygons?

The equilateral triangle has ugly irrational perimeter = area, but some of the other segments…

Concentrate on a side? an altitude? apothem? segment from center to vertex?

An interesting problem, dear readers, can we get an interesting problem out of this?

is up. It is the “rushed” edition, because it was thrown together quickly. But it doesn’t show.

There are a dozen links, including to some major sites, which submitted (the host solicited, which is what I do…).

The post about

is wonderful.

Over at his blog, Dave Marain asked for the fourth coordinate, along with (a,b), (0,0) and (b,a) to give a parallelogram. And then he asked for the area (in at least five ways!)

Later, Dave clarified that he wanted (a+b,a+b). IOW, the origin was to be the second coordinate. And, WLOG assuming b > a > 0, the area of the parallelogram (actually a rhombus) is .

However, without specifying the order of the coordinates, we could have used (a-b, b-a) for the missing coordinate. Can we find the area of this parallelogram?

Can we find the area of any triangle from just its coordinates (0,0), (a,b), (c,d) ?  Any arbitrary quadrilateral? I think translating to get one point at the origin is ok, but rotating is not…