Puzzle for November 19, 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 both B and D to each side of eq.2: A – B + B + D = C – D + B + D which becomes A + D = C + B which may be written as A + D = B + C In eq.3, replace B + C with A + D: D + F = A + D Subtract D from both sides of the equation above: D + F – D = A + D – D which makes F = A
Hint #2
In eq.3, replace C with B + D (from eq.6): D + F = B + B + D Subtract D from each side of the above equation: B + B + D – D = D + F – D which makes 2×B = F which also makes A = F = 2×B
Hint #3
In eq.5, substitute B + D for C (from eq.6), and 2×B for A and F: B + D + E = 2×B – B + D + 2×B which becomes B + D + E = 3×B + D Subtract both B and D from each side of the equation above: B + D + E – B – D = 3×B + D – B – D which simplifies to E = 2×B
Hint #4
Substitute 2×B for E, and B + D for C (from eq.6) in eq.4: 2×B – D = B + D + D In the above equation, add D to both sides, and subtract B from each side: 2×B – D + D – B = B + D + D + D – B which simplifies to B = 3×D Divide both sides by 3: B ÷ 3 = 3×D ÷ 3 which means ⅓×B = D
Hint #5
Substitute ⅓×B for D in eq.6: B + ⅓×B = C which means 1⅓×B = C
Solution
Substitute 2×B for A and E and F, 1⅓×B for C, and ⅓×B for D in eq.1: 2×B + B + 1⅓×B + ⅓×B + 2×B + 2×B = 26 which becomes 8⅔×B = 26 Divide both sides by 8⅔: 8⅔×B ÷ 8⅔ = 26 ÷ 8⅔ which means B = 3 making A = E = F = 2×B = 2 × 3 = 6 C = 1⅓×B = 1⅓ × 3 = 4 D = ⅓×B = ⅓ × 3 = 1 and ABCDEF = 634166