Puzzle for December 28, 2020 ( )
Scratchpad
Find the 6-digit number ABCDEF by solving the following equations:
A, B, C, D, E, and F each represent a one-digit positive integer.
Scratchpad
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Hint #1
Add D to both sides of eq.3: F – D + D = A + D + D which becomes eq.3a) F = A + 2×D In eq.1, replace F with A + 2×D (from eq.3a): A + B + D = A + 2×D Subtract A and D from each side of the equation above: A + B + D – A – D = A + 2×D – A – D which makes B = D
Hint #2
In eq.2, replace D with B: B + F = C + B Subtract B from each side of the above equation: B + F – B = C + B – B which makes F = C
Hint #3
In eq.4, substitute F for C: E – F = F + F which becomes E – F = 2×F Add F to both sides of the above equation: E – F + F = 2×F + F which makes E = 3×F
Hint #4
Substitute 3×F for E in eq.6: C × F = 3×F Divide both sides of the equation above by F: C × F ÷ F = 3×F ÷ F which makes C = 3 and makes F = C = 3 and also makes E = 3×F = 3×3 = 9
Hint #5
Substitute 3 for C, and 3 for F in eq.5: B × 3 = 3 Divide both sides of the above equation by 3: B × 3 ÷ 3 = 3 ÷ 3 which makes B = 1 and also makes D = B = 1
Solution
Substitute 3 for F, and 1 for D in eq.3a: 3 = A + 2×1 which becomes 3 = A + 2 Subtract 2 from each side of the equation above: 3 – 2 = A + 2 – 2 which makes 1 = A and makes ABCDEF = 113193