Puzzle for December 26, 2019 ( )
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 D and F to both sides of eq.5: C + D – F + D + F = A + B – (D + F) + D + F which becomes C + 2×D = A + B – D – F + D + F which becomes eq.5a) C + 2×D = A + B Add D to both sides of eq.3: A + B – D + D = D + E + D which becomes eq.3a) A + B = 2×D + E
Hint #2
In eq.5a, replace A + B with 2×D + E (from eq.3a): C + 2×D = 2×D + E Subtract 2×D from both sides of the equation above: C + 2×D – 2×D = 2×D + E – 2×D which makes C = E
Hint #3
Add B to both sides of eq.4: B + E + B = A – B + C + B which becomes 2×B + E = A + C In the equation above, replace C with E: 2×B + E = A + E Subtract E from each side of the equation above: 2×B + E – E = A + E – E which makes 2×B = A
Hint #4
In eq.2, replace C with E: E + F = B + E + D – F In the above equation, subtract E from each side, and add F to both sides: E + F – E + F = B + E + D – F – E + F which makes eq.2a) 2×F = B + D
Hint #5
Add (A – F) to both sides of eq.6: D – (A – F) + (A – F) = A – F + (A – F) which becomes eq.6a) D = 2×A – 2×F In eq.6a, substitute (2×B) for A, and (B + D) for 2×F (from eq.2a): D = 2×(2×B) – (B + D) which becomes D = 4×B – B – D which becomes D = 3×B – D Add D to each side of the equation above: D + D = 3×B – D + D which makes 2×D = 3×B Divide both sides by 2: 2×D ÷ 2 = 3×B ÷ 2 which makes D = 1½×B
Hint #6
Substitute 1½×B for D in eq.2a: 2×F = B + 1½×B which means 2×F = 2½×B Divide both sides of the equation above by 2: 2×F ÷ 2 = 2½×B ÷ 2 which makes F = 1¼×B
Hint #7
Substitute (1½×B) for D, and 2×B for A in eq.5a: C + 2×(1½×B) = 2×B + B which becomes C + 3×B = 3×B Subtract 3×B from both sides of the equation above: C + 3×B – 3×B = 3×B – 3×B which makes C = 0 and also makes E = C = 0
Solution
Substitute 2×B for A, 0 for C and E, 1½×B for D, and 1¼×B for F in eq.1: 2×B + B + 0 + 1½×B + 0 + 1¼×B = 23 which simplifies to 5¾×B = 23 Divide both sides of the equation above by 5¾: 5¾×B ÷ 5¾ = 23 ÷ 5¾ which means B = 4 making A = 2×B = 2 × 4 = 8 D = 1½×B = 1½ × 4 = 6 F = 1¼×B = 1¼ × 4 = 5 and ABCDEF = 840605