Puzzle for October 20, 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
Add E and F to each side of eq.6: D – E + E + F = C – F + E + F which becomes D + F = C + E In the equation above, replace C + E with A + D (from eq.4): D + F = A + D Subtract D from both sides: D + F – D = A + D – D which makes F = A
Hint #2
In eq.3, replace A with F: F = F + B Subtract F from each side of the equation above: F – F = F + B – F which makes 0 = B
Hint #3
In eq.2, substitute 0 for B: 0 + D = A which makes D = A
Hint #4
Substitute A for D in eq.4: A + A = C + E which becomes eq.4a) 2×A = C + E In eq.5, add C to both sides, and subtract E from both sides: C – E + C – E = A – C + E + C – E which becomes eq.5a) 2×C – 2×E = A
Hint #5
Substitute (2×C – 2×E) for A (from eq.5a) in eq.4a: 2×(2×C – 2×E) = C + E which becomes 4×C – 4×E = C + E In the equation above, add 4×E to each side, and subtract C from each side: 4×C – 4×E + 4×E – C = C + E + 4×E – C which becomes 3×C = 5×E Divide both sides by 3: 3×C ÷ 3 = 5×E ÷ 3 which becomes C = 1⅔×E
Hint #6
Substitute (1⅔×E) for C in eq.5a: 2×(1⅔×E) – 2×E = A which becomes 3⅓×E – 2×E = A which makes 1⅓×E = A and also makes D = A = F = 1⅓×E
Solution
Substitute 1⅓×E for A and D and F, 0 for B, and 1⅔×E for C in eq.1: 1⅓×E + 0 + 1⅔×E + 1⅓×E + E + 1⅓×E = 20 which simplifies to 6⅔×E = 20 Divide each side of the equation above by 6⅔: 6⅔×E ÷ 6⅔ = 20 ÷ 6⅔ which means E = 3 making A = D = F = 1⅓×E = 1⅓ × 3 = 4 C = 1⅔×E = 1⅔ × 3 = 5 and ABCDEF = 405434