Puzzle for December 15, 2021 ( )
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.6, replace E + F with B + D (from eq.3): B + C = D + B + D which becomes B + C = B + 2×D Subtract B from both sides of the above equation: B + C – B = B + 2×D – B which makes C = 2×D
Hint #2
Subtract F and B from each side of eq.2: D + F – F – B = A + B – F – B which becomes D – B = A – F In the above equation, replace D – B with B – A (from eq.5): B – A = A – F Add A and F to both sides of the equation above: B – A + A + F = A – F + A + F which becomes eq.2a) B + F = 2×A
Hint #3
Add A to both sides of eq.4: C – A + A = A – B + A which becomes C = 2×A – B In the above equation, substitute B + F for 2×A (from eq.2a): C = B + F – B which makes C = F and also makes eq.4a) C = F = 2×D
Hint #4
In eq.2, substitute 2×D for F (from eq.4a): D + 2×D = A + B which becomes 3×D = A + B Subtract B from each side of the equation above: 3×D – B = A + B – B which becomes eq.2b) 3×D – B = A
Hint #5
Substitute (3×D – B) for A (from eq.2b) in eq.5: B – (3×D – B) = D – B which becomes B – 3×D + B = D – B which becomes 2×B – 3×D = D – B Add B and 3×D to both sides of the above equation: 2×B – 3×D + B + 3×D = D – B + B + 3×D which makes 3×B = 4×D Divide both sides by 4: 3×B ÷ 4 = 4×D ÷ 4 which makes ¾×B = D
Hint #6
Substitute (¾×B) for D in eq.4a: C = F = 2×(¾×B) which makes C = F = 1½×B
Hint #7
Substitute (¾×B) for D in eq.2b: 3×(¾×B) – B = A which becomes 2¼×B – B = A which makes 1¼×B = A
Hint #8
Substitute 1½×B for F, and ¾×B for D in eq.3: E + 1½×B = B + ¾×B which becomes E + 1½×B = 1¾×B Subtract 1½×B from both sides of the equation above: E + 1½×B – 1½×B = 1¾×B – 1½×B which makes E = ¼×B
Solution
Substitute 1¼×B for A, 1½×B for C and F, ¾×B for D, and ¼×B for E in eq.1: 1¼×B + B + 1½×B + ¾×B + ¼×B + 1½×B = 25 which simplifies to 6¼×B = 25 Divide both sides of the above equation by 6¼: 6¼×B ÷ 6¼ = 25 ÷ 6¼ which means B = 4 making A = 1¼×B = 1¼ × 4 = 5 C = F = 1½×B = 1½ × 4 = 6 D = ¾×B = ¾ × 4 = 3 E = ¼×B = ¼ × 4 = 1 and ABCDEF = 546316