Puzzle for June 17, 2022 ( )
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: E + F = A + D + B In the equation above, replace A + D with B + E (from eq.4): E + F = B + E + B which becomes E + F = 2×B + E Subtract E from each side: E + F – E = 2×B + E – E which makes F = 2×B
Hint #2
In eq.5, replace D with B + C (from eq.2), and F with 2×B: C + E = B + C + 2×B which becomes C + E = 3×B + C Subtract C from each side of the equation above: C + E – C = 3×B + C – C which makes E = 3×B
Hint #3
Subtract the left and right sides of eq.5 from the left and right sides of eq.3, respectively: C + D – (C + E) = A + F – (D + F) which becomes C + D – C – E = A + F – D – F which becomes D – E = A – D Add E and D to both sides of the equation above: D – E + E + D = A – D + E + D which becomes eq.3a) 2×D = A + E
Hint #4
In eq.3a, substitute 3×B for E: eq.3b) 2×D = A + 3×B
Hint #5
In eq.4, substitute 3×B for E: B + 3×B = A + D which becomes 4×B = A + D Subtract D from each side of the equation above: 4×B – D = A + D – D which becomes eq.4a) 4×B – D = A
Hint #6
Substitute 4×B – D for A (from eq.4a) into eq.3b: 2×D = 4×B – D + 3×B which becomes 2×D = 7×B – D Add D to both sides of the above equation: 2×D + D = 7×B – D + D which makes 3×D = 7×B Divide both sides by 3: 3×D ÷ 3 = 7×B ÷ 3 which makes D = 2⅓×B
Hint #7
Substitute 2⅓×B for D in eq.4a: 4×B – 2⅓×B = A which makes 1⅔×B = A
Hint #8
Substitute 2⅓×B for D in eq.2: 2⅓×B = B + C Subtract B from each side of the above equation: 2⅓×B – B = B + C – B which makes 1⅓×B = C
Solution
Substitute 1⅔×B for A, 1⅓×B for C, 2⅓×B for D, 3×B for E, and 2×B for F in eq.1: 1⅔×B + B + 1⅓×B + 2⅓×B + 3×B + 2×B = 34 which simplifies to 11⅓×B = 34 Divide both sides of the above equation by 11⅓: 11⅓×B ÷ 11⅓ = 34 ÷ 11⅓ which makes B = 3 making A = 1⅔×B = 1⅔ × 3 = 5 C = 1⅓×B = 1⅓ × 3 = 4 D = 2⅓×B = 2⅓ × 3 = 7 E = 3×B = 3 × 3 = 9 F = 2×B = 2 × 3 = 6 and ABCDEF = 534796