Puzzle for March 19, 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.5: B – C + C + E = D – E + C + E which becomes eq.5a) B + E = D + C In eq.5a, replace B with D + E (from eq.2): D + E + E = D + C which becomes D + 2×E = D + C Subtract D from each side: D + 2×E – D = D + C – D which makes 2×E = C
Hint #2
In eq.3, replace C with 2×E: 2×E – A = E In the equation above, add A to both sides, and subtract E from each side: 2×E – A + A – E = E + A – E which makes E = A
Hint #3
In eq.6, substitute E for A, and 2×E for C: D + F = E + E + 2×E which becomes D + F = 4×E Subtract D from each side of the above equation: D + F – D = 4×E – D which becomes eq.6a) F = 4×E – D
Hint #4
Substitute 4×E – D for F (from eq.6a), and 2×E for C in eq.4: 4×E – D – D = B – 2×E which becomes 4×E – 2×D = B – 2×E Add 2×E to both sides of the above equation: 4×E – 2×D + 2×E = B – 2×E + 2×E which becomes eq.4a) 6×E – 2×D = B
Hint #5
Substitute 6×E – 2×D for B (from eq.4a), and 2×E for C in eq.5a: 6×E – 2×D + E = D + 2×E which becomes 7×E – 2×D = D + 2×E In the above equation, add 2×D to both sides, and subtract 2×E from each side: 7×E – 2×D + 2×D – 2×E = D + 2×E + 2×D – 2×E which simplifies to 5×E = 3×D Divide both sides by 3: 5×E ÷ 3 = 3×D ÷ 3 which makes 1⅔×E = D
Hint #6
Substitute (1⅔×E) for D in eq.4a: 6×E – 2×(1⅔×E) = B which becomes 6×E – 3⅓×E = B which makes 2⅔×E = B
Hint #7
Substitute 1⅔×E for D in eq.6a: F = 4×E – 1⅔×E which makes F = 2⅓×E
Solution
Substitute E for A, 2⅔×E for B, 2×E for C, 1⅔×E for D, and 2⅓×E for F in eq.1: E + 2⅔×E + 2×E + 1⅔×E + E + 2⅓×E = 32 which simplifies to 10⅔×E = 32 Divide both sides of the equation above by 10⅔: 10⅔×E ÷ 10⅔ = 32 ÷ 10⅔ which makes E = 3 making A = E = 3 B = 2⅔×E = 2⅔ × 3 = 8 C = 2×E = 2 × 3 = 6 D = 1⅔×E = 1⅔ × 3 = 5 F = 2⅓×E = 2⅓ × 3 = 7 and ABCDEF = 386537