Puzzle for July 8, 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
eq.6 may be written as: D = (A + C + F) ÷ 3 Multiply both sides of the above equation by 3: D × 3 = ((A + C + F) ÷ 3) × 3 which becomes eq.6a) 3×D = A + C + F
Hint #2
Add B and F to both sides of eq.5: A – B + C – D + B + F = B + E – F + B + F which becomes A + C – D + F = 2×B + E which may be written as A + C + F – D = 2×B + E In the above equation, replace A + C + F with 3×D (from eq.6a): 3×D – D = 2×B + E which becomes eq.5a) 2×D = 2×B + E
Hint #3
In eq.4, add A to both sides, and subtract D from both sides: B + D – A – E + A – D = C – D + F + A – D which becomes B – E = C – 2×D + F + A which may be written as B – E = A + C + F – 2×D In the above equation, replace A + C + F with 3×D (from eq.6a): B – E = 3×D – 2×D which becomes eq.4a) B – E = D
Hint #4
In eq.5a, substitute (B – E) for D (from eq.4a): 2×(B – E) = 2×B + E which becomes 2×B – 2×E = 2×B + E In the above equation, add 2×E to both sides, and subtract 2×B from both sides: 2×B – 2×E + 2×E – 2×B = 2×B + E + 2×E – 2×B which makes 0 = 3×E which means 0 = E
Hint #5
In eq.4a, replace E with 0: B – 0 = D which makes B = D
Hint #6
Substitute B for D, and 0 for E in eq.3: B + F – B = C + 0 – F which becomes F = C – F Add F to both sides of the equation above: F + F = C – F + F which makes 2×F = C
Hint #7
Substitute 2×F for C, B for D, and 0 for E in eq.2: 2×F + B + 0 = A + B – 0 which becomes 2×F + B = A + B Subtract B from each side of the above equation: 2×F + B – B = A + B – B which makes 2×F = A
Hint #8
Substitute 2×F for A and C in eq.6a: 3×D = 2×F + 2×F + F which becomes 3×D = 5×F Divide both sides of the above equation by 3: 3×D ÷ 3 = 5×F ÷ 3 which makes D = 1⅔×F and also makes B = D = 1⅔×F
Solution
Substitute 2×F for A and C, 1⅔×F for B and D, and 0 for E in eq.1: 2×F + 1⅔×F + 2×F + 1⅔×F + 0 + F = 25 which simplifies to 8⅓×F = 25 Divide both sides of the above equation by 8⅓: 8⅓×F ÷ 8⅓ = 25 ÷ 8⅓ which means F = 3 making A = C = 2×F = 2 × 3 = 6 B = D = 1⅔×F = 1⅔ × 3 = 5 and ABCDEF = 656503