Mathematical Entertainments by M.H. Greenblatt

Author:M.H. Greenblatt
Language: eng
Format: epub
Tags: Mathematics; Puzzles; Diophantine; Konigsberg; Euler; Fermat; Symmetry; Divisibility; Humour

Figure 23

of small black and white cubes. But the full 6X6X6 cube contains 14 X 8 cubes of one color, and 13 X 8 of the other. Hence, it cannot be formed of the 1 by 2 by 4 bricks.

Q.K.D.

9: Parlor Tricks and Number Manipulation

Eight 8*8

Can you take eight 8's and arrange them in any fashion connected by the usual signs of mathematics so that they represent exactly one thousand? This question appeared recently in a popular magazine. The big question appears to be what, exactly, do we mean by "the usual signs of mathematics"? Do we mean only the plus, minus, times, and divide signs? Some people allow the use of the square root sign a finite number of times; others would allow its use an infinite number of times. We can consider the use of the factorial function and the gamma function. The implication in the problem, as I originally saw it, was that there was only one solution. This is very false. (If ever a superlative adjective needed to be compared, this is it.) The solution they were thinking of was the following:

888 + 88 + 8 + 8 + 8 = 1000

A much simpler solution would have been:

8888 - 888

8

= M (1000)

Other solutions, which are only slightly more complex, are:

8 + 8

8

(8X8X8 8) -8 = M

82

Parlor Tricks and Number Manipulation S3

8(8X8 + 8X8)-8-8-8 = itf

; (8+8)8 _(!t±!±j?)] 8 = J ,

Still other solutions (which are equivalent to 10 5 ) are:

&*)

(8+8+8)/8

■ M and [8 +

g -L. g\(8+8+8)/8

{' + •-&

= M

If we use the factorial sign, we can get:

8!

.(.-»-±l±!)

-8 = M

~- - 81(8 X 8 X 8) - 8] - 8 - M

8! r_^+ju_ 8 = Af

|_8(88 - 8)J

If we now allow ourselves to use the gamma function, wc can get:

T(8) - (8 X 8)(s-8 - 0 - 8 = 8r(8)

M i = M

8+8+8+8+8 We can use logs (base 8) and get:

pog 8 8 X 8][8 X 8 X 8 - 8] - 8 = M

[10gSXSX8X8X8 8]r(8) - 8 = M

"(88 — 8}~|l ,0 S» (sxaxE

R88 - 8) 1'

[ 8 + ( !

g + 8\"|l lo & (8X8XB

= M

= M

Hi Parlor Tricks and Number Manipulation

And finally, we use the combinatorial symbol, where

On! = — f T-. and get: r\{n — r)l

•[en

-8

+ 8-8 = M

Possibly the most outrageous array of all is:

/88 \

( — X 8 X 8 X 8 J - 1000

Of course, this last isn't really legal because it represents one-zero-zero-zero in the number base 8. It looks like a thousand, but looks are often deceiving.

The Multiplication Theorem

The Multiplication Theorem of algebra, which is learned by all high school students (and forgotten by most), is very important for an understanding of many simple mental calculations. In one of its forms, it states that the product of two factors (a + 6) (c + d) is simply:

(o + 6)(e + d) = ac + ad + bc + bd (1)

The case where a = c and b = d is a special case which deserves some note. It represents (a + b) (a + b) or (o + by.

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