All about flooble | fun stuff | Get a free chatterbox | Free JavaScript | Avatars
 Click to listen to the single ▶Support album on Kickstarter perplexus dot info

 Closest to e (Posted on 2012-04-18)
This is in continuation of Closest to pi.

(A) As we are aware, the approximate value of e with some decimals is 2.7182818284590...

Write an expression using all digits (0-9) once each, to achieve the closest value to e. The digits 1, 2, 3, 4, 5, 6, 7, 8, 9, and 0 must be arranged strictly in this order. You can only use the five symbols: parentheses , addition, subtraction, multiplication and division symbols that is, (), +, -, *, /

Concatenating two or more digits is allowed. Each symbol can occur at most once in the given expression. Only standard order of operations is followed – that is, multiplication does not take precedence over division, addition does not take precedence over multiplication, and so on.

For example : (123+456)/7890 is a valid expression. However, expressions like 1+234/567 + 8/9 -0 or, 1+3*2456/78-90 are not allowed.

(B) Write an expression closest to e, if all the other conditions in (A) remain unaltered, but the restriction of each symbol occurring at most one is withdrawn. For example, expressions like 1+234/567 + 8/9 -0 is allowed. However, expressions like: 1+3*2456/78-90 are not allowed.

(C) What are the respective answers to (A) and (B) if a sixth symbol, that is, the decimal point is allowed?

 No Solution Yet Submitted by K Sengupta No Rating

Comments: ( Back to comment list | You must be logged in to post comments.)
 computer solution to A and B | Comment 4 of 6 |

A very minor tweak on the program used to solve the Close to Pi puzzle. This time run time was also speeded up by use of the QB64 compiler, thanks to brianjn's advocacy of that compiler.

DEFDBL A-Z
DIM SHARED stack(11)
DIM SHARED upTo, stackLevel
DIM SHARED h\$, best, usedPlus, usedMinus, usedTimes, usedDiv

best = 9999
stack(1) = 1: stackLevel = 1: upTo = 1
h\$ = "1"

OPEN "close2e.txt" FOR OUTPUT AS #2

CLOSE 2

nextOne = upTo + 1

SELECT CASE nextOne
CASE 10
nextOne = 0
CASE 1
' this is the end
' evaluate
IF stackLevel = 1 THEN
diff = ABS(stack(1) - 2.718281828#)
IF diff <= best THEN
best = diff
PRINT h\$, stack(1)
PRINT #2, h\$, stack(1)
END IF
EXIT SUB
ELSE
nextOne = 999
END IF
END SELECT

'first try concatenation
IF nextOne < 999 THEN
lc\$ = RIGHT\$(h\$, 1)
ix = INSTR("0123456789", lc\$)
IF ix THEN
h\$ = h\$ + LTRIM\$(STR\$(nextOne))
stack(stackLevel) = 10 * stack(stackLevel) + nextOne
utSave = upTo
upTo = nextOne
upTo = utSave
stack(stackLevel) = stack(stackLevel) \ 10
h\$ = LEFT\$(h\$, LEN(h\$) - 1)
END IF
END IF

'try comma
IF nextOne < 999 THEN
stackLevel = stackLevel + 1
stack(stackLevel) = nextOne
h\$ = h\$ + "," + LTRIM\$(STR\$(nextOne))
utSave = upTo
upTo = nextOne
upTo = utSave
stackLevel = stackLevel - 1
h\$ = LEFT\$(h\$, LEN(h\$) - 2)
END IF

EXIT SUB

IF stackLevel >= 2 THEN
s1 = stack(stackLevel - 1)
s2 = stack(stackLevel)
stackLevel = stackLevel - 1

stack(stackLevel) = s1 + s2
h\$ = h\$ + "+"
IF usedPlus = 0 THEN usedPlus = 1: addon: usedPlus = 0
h\$ = LEFT\$(h\$, LEN(h\$) - 1)

stack(stackLevel) = s1
stackLevel = stackLevel + 1
stack(stackLevel) = s2

s1 = stack(stackLevel - 1)
s2 = stack(stackLevel)
stackLevel = stackLevel - 1

stack(stackLevel) = s1 - s2
h\$ = h\$ + "-"
IF usedMinus = 0 THEN usedMinus = 1: addon: usedMinus = 0
h\$ = LEFT\$(h\$, LEN(h\$) - 1)

stack(stackLevel) = s1
stackLevel = stackLevel + 1
stack(stackLevel) = s2

s1 = stack(stackLevel - 1)
s2 = stack(stackLevel)
stackLevel = stackLevel - 1

stack(stackLevel) = s1 * s2
h\$ = h\$ + "*"
IF usedTimes = 0 THEN usedTimes = 1: addon: usedTimes = 0
h\$ = LEFT\$(h\$, LEN(h\$) - 1)

stack(stackLevel) = s1
stackLevel = stackLevel + 1
stack(stackLevel) = s2

s1 = stack(stackLevel - 1)
s2 = stack(stackLevel)

IF s2 <> 0 THEN
stackLevel = stackLevel - 1

stack(stackLevel) = s1 / s2
h\$ = h\$ + "/"
IF usedDiv = 0 THEN usedDiv = 1: addon: usedDiv = 0
h\$ = LEFT\$(h\$, LEN(h\$) - 1)

stack(stackLevel) = s1
stackLevel = stackLevel + 1
stack(stackLevel) = s2
END IF

END IF
RETURN

END SUB

finds the closest value without repeating an operation as:

1,234,56+/,789*,0-           2.720689655172414

where the value on the right is obtained by the operations on the left, which are shown in reverse Polish form. Translated into algebraic, that's

1/(234+56)*789-0 ~= 2.720689655172414 ,

where, it's to be remembered that multiplication does not take precedence over division, though this notation is ambiguous (see discussion at http://mathforum.org/library/drmath/view/57021.html).

If repetition is allowed:

1,2*,3/,4*,5,6,7,8/+,90+/+   2.718279569892473

where the RPN translates to 1*2/3*4+5/(6+7/8+90), which I would be more comfortable writing as

(1*2/3)*4+5/(6+7/8+90) ~= 2.718279569892473

to avoid the ambiguity of the relative precedence of multiplication and division.

If we also allow a leading unary negative sign:

1,2/,3-,4,5*,6,7+,8/,90+/-  -2.718281036834925

The equivalent of the RPN is 1/2-3-4*5/((6+7)/8+90), but since this is the approximation of negative, rather than positive, e, we negate to

-1/2+3+4*5/((6+7)/8+90) ~= 2.718281036834925.

 Posted by Charlie on 2012-04-18 16:13:17

 Search: Search body:
Forums (0)