## Tuesday, 24 January 2012

### Tuesday, 24 January 2012

Today is 24 day of the year.
24 = 2^3 x 3
24 = 11 + 13, i.e. the sum of two consecutive primes.
24 = 4 x 3 x 2 x 1 = 4! which means there are 24 ways of arranging the four letters a, b, c, and d.
24 is in two primitive, Pythagorean triplets (7, 24, 25) and (24, 143, 145).
24 is the eighth tribonacci number.
24 is the number of moves required to solve the a game of Frogs and Toads with four of each.

One definition of Alcuin's sequence is;
the nth member of the sequence, denoted a(n) = the number of triangles with integer sides and perimeter n.
Clearly n = 0, n = 1 and n = 2 have no solution.
n = 3 has the single solution of an equilateral triangle with sides of 1 unit.
n = 4 you would initially think has the solution 1, 1, 2 but a little further thought reveals this to be a straight line and not a triangle.
n = 5 has the single solution (1, 2, 2)
The first perimeter that gives rise to more than one solution is n = 7 with solutions of (1, 3, 3) and (2, 2, 3).
24 is 31 and 34 member of Alcuin's sequence.
There is a college named after Alcuin at the University of York.

The 24 solutions for 31 and 34 are:

1:    (1, 15, 15)   (2, 16, 16)
2:    (2, 14, 15)   (3, 15, 16)
3:    (3, 13, 15)   (4, 14, 16)
4:    (3, 14, 14)   (4, 15, 15)
5:    (4, 12, 15)   (5, 13, 16)
6:    (4, 13, 14)   (5, 14, 15)
7:    (5, 11, 15)   (6, 12, 16)
8:    (5, 12, 14)   (6, 13, 15)
9:    (5, 13, 13)   (6, 14, 14)
10:   (6, 10, 15)   (7, 11, 16)
11:   (6, 11, 14)   (7, 12, 15)
12:   (6, 12, 13)   (7, 13, 14)
13:   (7,  9, 15)   (8, 10, 16)
14:   (7, 10, 14)   (8, 11, 15)
15:   (7, 11, 13)   (8, 12, 14)
16:   (7, 12, 12)   (8, 13, 13)
17:   (8,  8, 15)   (9,  9, 16)
18:   (8,  9, 14)   (9, 10, 15)
19:   (8, 10, 13)   (9, 11, 14)
20:   (8, 11, 12)   (9, 12, 13)
21:   (9,  9, 13)  (10, 10, 14)
22:   (9, 10, 12)  (10, 11, 13)
23:   (9, 11, 11)  (10, 12, 12)
24:  (10, 10, 11)  (11, 11, 12)