Puzzle for May 15, 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 C and E to both sides of eq.3: B – C + C + E = C – E + C + E which becomes eq.3a) B + E = 2×C Add B, C, and E to both sides of eq.6: B – C + D – E + B + C + E = A – B + C + B + C + E which becomes eq.6a) 2×B + D = A + 2×C + E
Hint #2
In eq.6a, replace 2×C with B + E (from eq.3a): 2×B + D = A + B + E + E which becomes 2×B + D = A + B + 2×E Subtract B from both sides of the above equation: 2×B + D – B = A + B + 2×E – B which becomes eq.6b) B + D = A + 2×E
Hint #3
Add E and F to both sides of eq.4: B + D – E + E + F = A + E – F + E + F which becomes B + D + F = A + 2×E In the equation above, substitute B + D for A + 2×E (from eq.6b): B + D + F = B + D Subtract B and D from both sides: B + D + F – B – D = B + D – B – D which means F = 0
Hint #4
Substitute 0 for F in eq.2: B = D + 0 which means B = D
Hint #5
Substitute 0 for F in eq.5: A – 0 = C – A Add A to each side of the equation above: A – 0 + A = C – A + A which makes 2×A = C
Hint #6
Substitute (2×A) for C in eq.3a: B + E = 2×(2×A) which becomes B + E = 4×A Subtract E from each side of the above equation: B + E – E = 4×A – E which makes B = 4×A – E and also makes eq.3b) D = B = 4×A – E
Hint #7
Substitute 4×A – E for B and D (from eq.3b) in eq.6b: 4×A – E + 4×A – E = A + 2×E which becomes 8×A – 2×E = A + 2×E In the above equation, add 2×E to both sides, and subtract A from both sides: 8×A – 2×E + 2×E – A = A + 2×E + 2×E – A which makes 7×A = 4×E Divide each side by 4: 7×A ÷ 4 = 4×E ÷ 4 which makes 1¾×A = E
Hint #8
Substitute 1¾×A for E in eq.3b: D = B = 4×A – 1¾×A which makes eq.3c) D = B = 2¼×A
Solution
Substitute 2¼×A for B and D, 2×A for C, 1¾×A for E, and 0 for F in eq.1: A + 2¼×A + 2×A + 2¼×A + 1¾×A + 0 = 37 which simplifies to 9¼×A = 37 Divide both sides of the above equation by 9¼: 9¼×A ÷ 9¼ = 37 ÷ 9¼ which means A = 4 making B = D = 2¼×A = 2¼ × 4 = 9 C = 2×A = 2 × 4 = 8 E = 1¾×A = 1¾ × 4 = 7 and ABCDEF = 498970