Puzzle for September 1, 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
In eq.5, replace F with A + B (from eq.4): A – B = A + B – D In the equation above, subtract A from both sides: A – B – A = A + B – D – A which becomes –B = B – D Add B and D to both sides: –B + B + D = B – D + B + D which becomes D = 2×B
Hint #2
In eq.3, replace F with A + B (from eq.4): B + C = A + B – E In the above equation, subtract B from each side, and add E to each side: B + C – B + E = A + B – E – B + E which becomes eq.3a) C + E = A
Hint #3
In eq.2, substitute C + E for A (from eq.3a): D + E = C + E Subtract E from each side of the equation above: D + E – E = C + E – E which makes D = C and also makes eq.2a) D = C = 2×B
Hint #4
Substitute 2×B for D in eq.6: 2×B – E = B + E In the above equation, add E to both sides, and subtract B from both sides: 2×B – E + E – B = B + E + E – B which makes B = 2×E
Hint #5
Substitute (2×E) for B in eq.2a: D = C = 2×(2×E) which makes D = C = 4×E
Hint #6
Substitute 4×E for C in eq.3a: 4×E + E = A which makes 5×E = A
Hint #7
Substitute 5×E for A, and 2×E for B in eq.4: 5×E + 2×E = F which makes 7×E = F
Solution
Substitute 5×E for A, 2×E for B, 4×E for C and D, and 7×E for F in eq.1: 5×E + 2×E + 4×E + 4×E + E + 7×E = 23 which simplifies to 23×E = 23 Divide both sides of the above equation by 23: 23×E ÷ 23 = 23 ÷ 23 which means E = 1 making A = 5×E = 5 × 1 = 5 B = 2×E = 2 × 1 = 2 C = D = 4×E = 4 × 1 = 4 F = 7×E = 7 × 1 = 7 and ABCDEF = 524417