A/B*C+D*E*F/G+H*I=100
Use 1-9 for A-I. Use each number only once, and use all nine to make the equation true.
9/3*8+4*5*7/2+6*1=100
First, let’s group by evaluation order:
(A/B*C) + (D*E*F/G) + (H*I) = 100
Now let’s introduce some temporary variables:
X = (A/B*C)
Y = (D*E*F/G)
Z = (H*I)
So we get:
X + Y + Z = 100
We need to isolate a variable, so we rearrange:
X = 100 - (Y + Z)
Next, we are supposed to use the numbers 1 through 9 to make this true:
So Sum(A..I) = Sum(1..9) = 1+2+3+4+5+6+7+8+9 = (1+9) + (2+8) + (3+7) + (4+6) + 5
This reduces to 4(10)+5 or 45.
So because A + B + C + D + E + F + H + I = 45 we can say that for any particular letter s the following is true:
s = 45 - Sum({A..I}-s) where s is some letter A..I and {A..I}-s is the set A..I excluding itself.
Let’s take A. A = 45 - (B + C + D + E + F + H + I)
also
A = (X/C)*B
Using substitution on the second Equation we get:
A = [100 - (Y + Z)]/C*B
So
[100 - ( Y + Z )]/C*B = 45 - (B + C + D + E + F + H + I) = 45 - Sum(B..I)
[100 - ( Y + Z )]/C = [45 - Sum(B..I)]/B
100 - ( Y + Z ) = C*[45 - Sum(B..I)]/B
-Y -Z = C*[45 - Sum(B..I)]/B - 100
-Z = (C*[45 - Sum(B..I)]/B - 100)+Y
Z=-(C*[45 - Sum(B..I)]/B - 100)-Y
So, by substitutions
H*I =-(C*[45 - Sum(B..I)]/B - 100)-(D*E*F/G)
H={-(C*[45 - Sum(B..I)]/B - 100)-(D*E*F/G)}/I
Note now that this H has no A in it’s equality so we can substitute it out of one of the A equations:
A = 45 - (B + C + D + E + F + {-(C*[45 - Sum(B..I)]/B - 100)-(D*E*F/G)}/I + I)
And we continue with such variable-elimination until we get things down to manageable terms and then back substitute.
Or we could do it programmatically, here’s an algorithm [in Ada] to solve the equation:
With Ada.Text_IO;
Use Ada.Text_IO;
Procedure Solve_It is
-- Brute-force method
SubType Digit_Type is Positive range 1..9;
SubType Variable_Type is Character range 'A'..'I';
-- This type defines an array to some variable with some value.
Type Association_Type is Array(Variable_Type) of Digit_Type;
Procedure Display( Association : in Association_Type) is
Tab : Constant Character:= Character'Val(9);
begin
For Index in Association'Range loop
Put( Digit_Type'Image(Association(Index)) & Tab );
end loop;
Put_Line( "" );
end Display;
--This type allows us to have dynamic arrays
Type Association_Access is Access Association_Type;
-- This type defines an set of booleans to tell what has been used.
Type Set_Type is Array(Digit_Type) of Boolean;
Function Full( Set : Set_Type ) Return Boolean is
begin
Return Result: Boolean:= True do
For Index in Set'Range Loop -- Loop through the set and find
Result:= Result AND Set(Index); -- if any values are true, if so
end loop;
end return;
end Full;
Function Problem( State : Access Association_Type ) return Boolean is
-- Rename the variables for clarity.
A : Digit_Type:= State.All('A');
B : Digit_Type:= State.All('B');
C : Digit_Type:= State.All('C');
D : Digit_Type:= State.All('D');
E : Digit_Type:= State.All('E');
F : Digit_Type:= State.All('F');
G : Digit_Type:= State.All('G');
H : Digit_Type:= State.All('H');
I : Digit_Type:= State.All('I');
begin
-- Se if the equation is satisfied
Return 100 = A/B*C+D*E*F/G+H*I;
end Problem;
begin
For Index in Variable_Type'Range loop
Put( Character'Image(Index) & Character'Val(9) );
end loop;
Put_Line( "" );
For Index_A in Digit_Type'Range loop
For Index_B in Digit_Type'Range loop
For Index_C in Digit_Type'Range loop
For Index_D in Digit_Type'Range loop
For Index_E in Digit_Type'Range loop
For Index_F in Digit_Type'Range loop
For Index_G in Digit_Type'Range loop
For Index_H in Digit_Type'Range loop
For Index_I in Digit_Type'Range loop
Declare
State : Aliased Association_Type:=
Association_Type'(
'A' => INDEX_A, 'B' => INDEX_B, 'C' => INDEX_C, 'D' => INDEX_D,
'E' => INDEX_E, 'F' => INDEX_F, 'G' => INDEX_G, 'H' => INDEX_H,
'I' => INDEX_I
);
Fixed : Set_Type:= (Others => False);
Begin
For Index in State'Range loop
Fixed( State(Index) ):= True;
end loop;
if Full(Fixed) and then Problem(State'Access) then
Display( State );
end if;
End;
end loop;
end loop;
end loop;
end loop;
end loop;
end loop;
end loop;
end loop;
end loop;
end Solve_It;
The output is: 'A' 'B' 'C' 'D' 'E' 'F' 'G' 'H' 'I' 1 2 8 4 5 7 3 6 9 1 2 8 4 5 7 3 9 6 1 2 8 4 7 5 3 6 9 1 2 8 4 7 5 3 9 6 1 2 8 5 4 7 3 6 9 1 2 8 5 4 7 3 9 6 1 2 8 5 7 4 3 6 9 1 2 8 5 7 4 3 9 6 1 2 8 7 4 5 3 6 9 1 2 8 7 4 5 3 9 6 1 2 8 7 5 4 3 6 9 1 2 8 7 5 4 3 9 6 1 3 4 6 8 9 5 2 7 1 3 4 6 8 9 5 7 2 1 3 4 6 9 8 5 2 7 1 3 4 6 9 8 5 7 2 1 3 4 8 6 9 5 2 7 1 3 4 8 6 9 5 7 2 1 3 4 8 9 6 5 2 7 1 3 4 8 9 6 5 7 2 1 3 4 9 6 8 5 2 7 1 3 4 9 6 8 5 7 2 1 3 4 9 8 6 5 2 7 1 3 4 9 8 6 5 7 2 1 4 3 6 8 9 5 2 7 1 4 3 6 8 9 5 7 2 1 4 3 6 9 8 5 2 7 1 4 3 6 9 8 5 7 2 1 4 3 8 6 9 5 2 7 1 4 3 8 6 9 5 7 2 1 4 3 8 9 6 5 2 7 1 4 3 8 9 6 5 7 2 1 4 3 9 6 8 5 2 7 1 4 3 9 6 8 5 7 2 1 4 3 9 8 6 5 2 7 1 4 3 9 8 6 5 7 2 1 4 5 2 6 7 3 8 9 1 4 5 2 6 7 3 9 8 1 4 5 2 7 6 3 8 9 1 4 5 2 7 6 3 9 8 1 4 5 6 2 7 3 8 9 1 4 5 6 2 7 3 9 8 1 4 5 6 7 2 3 8 9 1 4 5 6 7 2 3 9 8 1 4 5 7 2 6 3 8 9 1 4 5 7 2 6 3 9 8 1 4 5 7 6 2 3 8 9 1 4 5 7 6 2 3 9 8 1 4 9 3 5 7 2 6 8 1 4 9 3 5 7 2 8 6 1 4 9 3 7 5 2 6 8 1 4 9 3 7 5 2 8 6 1 4 9 5 3 7 2 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8 1 7 6 4 2 3 5 9 8 1 7 6 4 2 5 3 9 8 6 3 4 7 1 2 5 9 8 6 3 4 7 1 5 2 9 8 6 3 7 4 1 2 5 9 8 6 3 7 4 1 5 2 9 8 6 4 3 7 1 2 5 9 8 6 4 3 7 1 5 2 9 8 6 4 7 3 1 2 5 9 8 6 4 7 3 1 5 2 9 8 6 7 3 4 1 2 5 9 8 6 7 3 4 1 5 2 9 8 6 7 4 3 1 2 5 9 8 6 7 4 3 1 5 2
Took me till midnight last night and half the morning, but I got it!
(6/2)*9+(5*7*8)/4+(3*1)=100
(Parentheses added for easier reading)
I’m one of those people who, if I see a puzzle, I can’t let it go until I solve it. There are some I’ve been mulling over for years looking for the answer. This one was nice, it fit on a single page ;)
Under the Rules of Affirmative action I have an answer:
1/2*3+4*5*6/7+8*9 = 100 !
I’m a black farmer and/or small businessWOMAN for the purposes of giving this answer.
In other words, you MUST accept it as a correct answer or be fined and.or jailed.