Search
Support DFF
If you benefit from the website, in terms of
knowledge, entertainment value, or something otherwise useful,
consider making a donation via PayPal to help defray the
costs. (No PayPal account necessary to donate via credit
card.) Transaction is secure.
If you shop at Amazon anyway,
consider using this link. We receive a few cents from each purchase.
Thanks.
Contact
Feedback:
Send an e-mail with your
comments about this program (or anything else).
|
| |
Problem Description
I recently received this email:
Dear Delphiforfun,
I have been investigating some variations on a classical problem involving
pandigital fractions and wonder if you have seen solutions.
Classical Problem -- I solved this one
If each of the first 9 letters represents a different number--1 through 9, the
smallest integer represented by: A/BC + D/EF + G/HI = 1; this assumes that the
denominators are two-digit numbers and NOT multiplied.
My Variations -- Need Verification of Solutions
I have modified the classical equation so that the denominators are multiplied
as follows: A/(B*C) + D/(E*F) + G/(H*I) and believe the smallest integral answer
is 2. Can you verify that?
For (A/B)*C + (D/E)*F + (G/H)*I, I think the smallest integral answer is 5.
For (A/B)^C + (D/E)^F + (G/H)^I. the smallest integral answer I have found is
1,100, but I suspect there is a smaller solution.
Regards, Jerry
Well Jerry, assuming that you are not a programmer and were doing this by
hand,
1 correct out of 3 isn't bad!
Background & Techniques
I wrote the simple code to solve these 4 problems (the classical version and
the three variations).. I emailed the results back to Jerry, but so
far, have received no response. Maybe he took it hard that two of
his three answers were incorrect.
Each of the four uses the same technique, namely:
Generate all permutations of the digits 1 through 9. For each permutation,
assign the digits of the result in order to the 9 variables A - I and evalua6te
the expression. We'll do the evaluation in extended mode to retain
fractional information. We check that the result is and integer and if so,
that it is a new minimum before adding it to the potential solutions
list.
The GetNext function from our old standby, the Combo
unit, returns the 360,000 or so permutations of the 9 digits. Any rational
numbers whose decimal representation is an infinite repeating non-zero
string cannot be represented exactly in floating point form. To get
around this problem, we look for values that are very close to an integer
(10-32 in this case). This test and the new minimum
test are incorporated into procedure CheckAndShowValues. Procedure AssignValues
assigns the permutation output to the variables. So each solution search
contains a six statement loop like this, with only the "x:=.."
statement changing :
while combos.getnext do
begin
assignvalues;
x:=a/(10*b+c)+d/(10*e+f)+g/(10*h+i); {for example}
CheckAndShowValues(x);
end;
Running/Exploring the Program
Suggestions for Further Explorations
 |
You can
probably think up other expression variations to evaluate. |
|
The
expressions could be selected in a radio box and a single button used to
search (with a case statement to select the
expression). |
| Created: July 24, 2003 |
Modified: May 18, 2009
|
|
