Puzzle for January 30, 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 non-negative integer.
Scratchpad
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Hint #1
eq.6 may be written as: A + D + C = B + E In the above equation, replace A + D with B (from eq.2): B + C = B + E Subtract B from both sides: B + C – B = B + E – B which makes C = E
Hint #2
In eq.4, replace B with A + D (from eq.2): D + F = A + A + D which becomes D + F = 2×A + D Subtract D from both sides of the equation above: D + F – D = 2×A + D – D which makes F = 2×A
Hint #3
In eq.5, substitute E for C, and 2×A for F: E – D = A + 2×A which becomes eq.5a) E – D = 3×A
Hint #4
In eq.3, substitute 2×A for F: E + 2×A = B + D – 2×A Subtract 2×A and D from both sides of the above equation: E + 2×A – 2×A – D = B + D – 2×A – 2×A – D which becomes eq.3a) E – D = B – 4×A
Hint #5
In eq.5a, substitute B – 4×A for E – D (from eq.3a): B – 4×A = 3×A Add 4×A to both sides of the above equation: B – 4×A + 4×A = 3×A + 4×A which makes B = 7×A
Hint #6
Substitute 7×A for B in eq.2: 7×A = A + D Subtract A from both sides of the equation above: 7×A – A = A + D – A which makes 6×A = D
Hint #7
Substitute 6×A for D in eq.5a: E – 6×A = 3×A Add 6×A to both sides: E – 6×A + 6×A = 3×A + 6×A which makes E = 9×A and also makes C = E = 9×A
Solution
Substitute 7×A for B, 9×A for C and E, 6×A for D, and 2×A for F in eq.1: A + 7×A + 9×A + 6×A + 9×A + 2×A = 34 which simplifies to 34×A = 34 Divide both sides of the above equation by 34: 34×A ÷ 34 = 34 ÷ 34 which means A = 1 making B = 7×A = 7 × 1 = 7 C = E = 9×A = 9 × 1 = 9 D = 6×A = 6 × 1 = 6 F = 2×A = 2 × 1 = 2 and ABCDEF = 179692