Pizza Slices

Figure 1

Take a pizza and pick an arbitrary point in it. Suppose you cut the pizza into 8 slices by cutting at 45 degree angles through that point, and color the alternate pieces red and green.

Surprising theorem: the total area of the red slices and the total area of the green slices will always be the same!

In fact, this theorem is true if the number of slices is any multiple of 4 except for 4, and the slices are cut by using equal angles through a fixed arbitrary point in the pizza.

Alternatively, if instead of equal angles, you use equal-length arcs on the circumference and slice from a fixed arbitrary point in the pizza, the conclusion still holds if the number of slices is even and greater than 2.

Interesting and Little-Known Algebra and Geometry Facts

Here are a few helpful and neat little facts that evade most students and teachers of algebra and geometry:

a) Though everyone can factor a^2 - b^2, a3 - b^3, a^3 + b^3 and a^4 - b^4, most folks do not know that:

a^4 + b^4 = (a^2 + ab(sqrt(2)) + b^2) (a^2 - ab(sqrt(2)) + b^2

b) An approximation exists for the factorial function (for large n) which seems hardly related but works:

Stirling's Formula ---> n! ~~ e^(-n) n^n sqrt(2 (pi) n)

c) The area of any regular polygon of n sides, each of length x, is given by:

Area = (1/4)nx^2 (cot(180°/n))

d) The radii of circumscribed (R) and inscribed (r) circles within such regular polygons are given by:

R = (x/2) csc (180°/n) and r = (x/2) cot (180°/n)

e) The radius of a circle inscribed within any triangle of sides a, b, and c with semi-perimeter s is given by:

r = (sqrt (s (s-a) (s-b) (s-c)) / s

e) The radius of a circle circumscribed about any triangle of sides a, b, and c with semi-perimeter s is given by:

R = abc / 4 (sqrt (s (s-a) (s-b) (s-c))

f) The perimeter P and area A of polygons (of n sides) inscribed in a circle of radius r is given by:

P = 2nr sin(pi/n) and A = (1/2) nr^2 sin (2pi/n)

g) The perimeter P and area A of polygons (of n sides) circumscribed about a circle of radius r is given by:

P = 2nr tan (pi/n) and A = nr^2 tan (pi/n)

h) Factoring the seemingly prime expression a^4 + 4b^4 becomes (and there are a family of these)

a^4 + 4b^4 = (a^2 + 2b^2)^2 - (2ab)^2 (thanks to N. Hobson)

500 Digits of e

Named after the world famous mathematician and extreme child prodigy Leonhard Euler, the natural logarithmic base has innumerable applications in all fields of science, business, and mathematics...here is just the first 500 digits or so...

2.71828 18284 59045 23536 02874 71352 66249
77572 47093 69995 95749 66967 62772 40766 30353 54759 45713
82178 52516 64274 27466 39193 20030 59921 81741 35966 29043
57290 03342 95260 59563 07381 32328 62794 34907 63233 82988
07531 95251 01901 15738 34187 93070 21540 89149 93488 41675
09244 76146 06680 82264 80016 84774 11853 74234 54424 37107
53907 77449 92069 55170 27618 38606 26133 13845 83000 75204
49338 26560 29760 67371 13200 70932 87091 27443 74704 72306
96977 20931 01416 92836 81902 55151 08657 46377 21112 52389
78442 50569 53696 77078 54499 69967 94686 44549 05987 93163
68892 30098 79312 77361 78215 42499 92295 76351 48220 82698
95193 66803 31825 28869 39849 64651 05820 93923 98294 88793
32036 25094 43117 30123 81970 68416 14039 70198 37679 32068
32823 76464 80429 53118 02328 78250 98194 55815 30175 67173

Chisenbop Multiplying by 9

Hold out your hands in front of you so that your thumbs point toward one another.

Visualize that your left pinky finger represents 1, the next finger 2, and so on left to right, until your right pinky finger represents 10. Those fingers represent the number you wish to multiply by 9. To do so, simply put the finger down you wish to multiply by 9. All fingers to the left of the down finger represent the tens digit of the answer while all fingers to the right represent the ones digit.

Example: 6 x 9. Put the finger representing 6 down (the right hand thumb). To the left of the down finger, you have 5 fingers up. That's your tens digit, 5. To the right, you have 4 fingers up. There's your ones digit, 4. Put those together and you have your answer: 54. Pretty cute.

Arithmetic Curiosities

- Here are just a few interesting patterns in arithmetic that you or your students may explore. Verify these results with paper and pencil or with calculator (if you must):

1 x 9 + 2 = 11	9 x 9 + 7 = 88	9 x 9 = 81	6 x 7 = 42
12 x 9 + 3 = 111	98 x 9 + 6 = 888	99 x 99 = 9801	66 x 67 = 4422
123 x 9 + 4 = 1111	987 x 9 + 5 = 8888	999 x 999 = 998001	666 x 667 = 444222
1234 x 9 + 3 = 11111	9876 x 9 + 4 = 88888	9999 x 9999 = 99980001	6666 x 6667 = 44442222
12345 x 9 + 6 = 111111 123456 x 9 + 7 = 1111111 1234567 x 9 + 8 = 11111111 12345678 x 9 + 9 = 111111111 123456789 x 9 +10= 1111111111	1 x 8 + 1 = 9 12 x 8 + 2 = 98 123 x 8 + 3 = 987 1234 x 8 + 4 = 9876 12345 x 8 + 5 = 98765 123456 x 8 + 6 = 987654 1234567 x 8 + 7 = 9876543 12345678 x 8 + 8 = 98765432 123456789 x 8 + 9 = 987654321	9 x 9 + 7 = 88 98 x 9 + 6 = 888 987 x 9 + 5 = 8888 9876 x 9 + 4 = 88888 98765 x 9 + 3 = 888888 987654 x 9 + 2 = 8888888 9876543 x 9 + 1 = 88888888 98765432 x 9 + 0 = 888888888	1 x 1 = 1 11 x 11 = 121 111 x 111 = 12321 1111 x 1111 = 1234321 11111 x 11111 = 123454321 111111 x 111111 = 12345654321 1111111 x 1111111 = 1234567654321 11111111 x 11111111 = 123456787654321 111111111 x 111111111= 12345678987654321