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MATHEMATICAL WATCHES
Teresa Efigenia Alarcón Martínez*, Iván Guillén Escamilla **
*Departamento de Ciencias Computacionales e Ingenierías
** Departamento de Ciencias Naturales y Exactas
Centro Universitario de los Valles
Universidad de Guadalajara.
This section is dedicated to answer the problems related with the watches, in which the
numbers are expressed using a mathematical constant or a mathematical expression.
From the pages http://simplementenumeros.blogspot.com/2011/07/733-relojesmatematicos.html and
http://www.google.com.mx/search?q=relojes+matem%C3%A1ticos&hl=es419&client=firefox-a&hs=BLr&rls=org.mozilla:enUS:official&prmd=imvns&tbm=isch&tbo=u&source=univ&sa=X&ei=FNCNTpSBH
MassAK2hMGIAQ&ved=0CEoQsAQ&biw=1831&bih=959
were selected 5 images with mathematical watches.
Watch No. 1
FIGURE 1. Numbers are expressed using combinatorial expressions, factorial, rounding, mathematical
constants, numerical representations using a base different to decimal and html code.
http://simplementenumeros.blogspot.com/2011/07/733-relojes-matematicos.html
The explanation for the watch in Figure 1 appears in the Table No. 1
Number
TABLE 1. Answers for the watch No. 1.
Mathematical expression
1
2
Meaning
is the Legendre's constant
that is a mathematical constant
occurring in a formula
conjectured by Adrien-Marie
Legendre to capture
the asymptotic behavior of
the prime-counting
function
Its value is now
known to be exactly 1 [1].
∑
This is a geometric serie of
ratio ,
∑
∑( )
3
&#33
This is the number 3 in HTML
code. 33 is the hexadecimal
representation of the number
51 and 51 is the ASCII code of
number 3.
4
20 mod 8
4 is the rest of the division
between 20 and 8.
5
(
6
3!
7
)
is the golden ratio [2], and it
is equal to
)
.
√
, so (
The factorial of 3 is equal 6.
̅
7 is the number
rounding.
̅
after
8
9
The first ball in the figure is on
(active), the rest ones are off
(inactive).
The
binary
representation for 8 is 1000 in
which only the first bit is equal
to 1 (is active).
214
The number 9 using base 4 is
21 = 4*2+1.
(( ))
( )
11
0x0B
The number 11 in hexadecimal
representation is B.
12
√
12 is the cube root of 1728.
10
Watch No. 2
FIGURE 2. Numbers are expressed using different algebraical expressions.
The explanation for the watch in Figure 2 appears in the Table No. 2
Number
Mathematical
expression
1
2
TABLE 2. Answers to the watch No 2.
Meaning
The rest of the numbers is equal 1.
√
The square root of 4 is 2.
3
3 is quotient .
4
4 is the x value.
5
5 is the quotient.
6
6 is the final value of the expression.
7
52-x2+x=10
The roots of the polynomial -x2+x+42 are 7 and -6. For this
case only one solution is considered: 7.
8
√
8 is the square root of 64.
9
(
)
The result of the arithmetic operation (
Finally the product is 9.
10
-8=2-x
10 is the value of x in the equation.
11
1221:11121
11 is the quotient .
12
6*2
12 is the product of 6 and 2.
) is rounded to 3.
Watch No. 3
FIGURE 3. Numbers are expressed using different mathematical expressions.
The explanation for the watch in Figure 3 appears in the Table No. 3.
Number
Mathematical
expression
TABLE 3. Answers to the watch No 3.
Meaning
1
. The value of the expression is -1. The
appositive value is 1.
2
∑
This is a geometric serie ,of ratio
∑
∑( )
,
3
⌊ ⌋
3 nearest integer .
4
5
4 is the value of 2+2.
(
)
6
( )
7
̅
is the golden ratio, and it is equal to
√
, so (
)
.
The recursion formula for the Gamma function is:
(
)
( ). If n is nonnegative integer, i. e, 0,1, 2, …,
then:
(
)
. For n = 3. The factorial of 3 is 6.
7 is the rounded number.
8
is the Legendre's constant that is a mathematical constant. Its
value is known to be exactly 1 [1]. After adding 1 we obtain 8.
9
|
10
119
10 on base 9 is represented by 11.
11
√
The cube root of 1331 is 11.
|
The expression denotes the determinant of the matrix (
12
The
(
)
(
)
).
. The final values of the expression is 12.
Watch No. 4
FIGURE 4. Numbers are expressed using binary numeral system.
All numbers in the watch No. 4 are represented using binary numeral system.
Watch No. 5
FIGURE 5. Numbers are expressed using different mathematical expressions and constants.
The explanation for the watch in Figure 5 appears in the Table No. 4
Number
Mathematical
expression
1
TABLE 4. Answers to the watch No 5.
Meaning
The tangent of 45 grades is equal to 1.
2
√
The square root of 4 is 2.
3
Round( )
3 is the nearest integer to .
4
(
)
The logarithm of 55 to base e is equal to 4.0073331852…, 4 is
the rounded number.
5
The value of x is 5.
6
The factorial of 3 is 6.
7
̅
The nearest integer to
̅ is 7.
8
The quotient is 8.
9
3x3
10
gn
11
12
The product is 9.
The standard acceleration of gravity [3] is 9.806 65 m·s−2 . The
rounded value is 10.
is the number 11 using hexadecimal numeral representation.
is the number 12 using binary numeral representation.
REFERENCES
[1]. Legendre, A. M. Essai sur la Théorie des Nombres. Paris: Duprat, 1808.
[2]. Livio, Mario. The Golden Ratio: The Story of Phi, The World´s Most Astonishing Number, New York:
Broadway Books, 2002.
[3]. Métodos matemáticos para físicos, George Brown Arfken, Hans J. Weber; 5ta Edicion, Harcourt Academic
Press, 2001.