Puzzle for June 3, 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
Subtract the left and right sides of eq.3 from the left and right sides from eq.5, respectively: D + F – (F – D) = B + E – C – (E – C) which becomes D + F – F + D = B + E – C – E + C which makes 2×D = B
Hint #2
In eq.2, replace B with 2×D: A = 2×D + D which makes A = 3×D
Hint #3
In eq.6, replace A with 3×D, and B with 2×D: C + D = 3×D + 2×D – E which becomes C + D = 5×D – E In the equation above, subtract D from both sides, and add E to both sides: C + D – D + E = 5×D – E – D + E which becomes eq.6a) C + E = 4×D
Hint #4
In eq.4, substitute 3×D for A: E + F = 3×D + D which becomes eq.4a) E + F = 4×D
Hint #5
Substitute C + E for 4×D (from eq.6a) into eq.4a: E + F = C + E Subtract E from each side of the above equation: E + F – E = C + E – E which makes F = C
Hint #6
Add D and C to both sides of eq.3: F – D + D + C = E – C + D + C which becomes F + C = E + D Substitute C for F in the above equation: C + C = E + D which becomes 2×C = E + D which may be written as eq.3a) 2×C = D + E
Hint #7
Add E to both sides of eq.6: C + D + E = A + B – E + E which becomes C + D + E = A + B Substitute 2×C for D + E (from eq.3a), 3×D for A, and 2×D for B in the above equation: C + 2×C = 3×D + 2×D which makes 3×C = 5×D Divide both sides by 3: 3×C ÷ 3 = 5×D ÷ 3 which makes C = 1⅔×D and also makes F = C = 1⅔×D
Hint #8
Substitute (1⅔×D) for C in eq.3a: 2×(1⅔×D) = D + E which becomes 3⅓×D = D + E Subtract D from each side of the equation above: 3⅓×D – D = D + E – D which becomes 2⅓×D = E
Solution
Substitute 3×D for A, 2×D for B, 1⅔×D for C and F, and 2⅓×D for E in eq.1: 3×D + 2×D + 1⅔×D + D + 2⅓×D + 1⅔×D = 35 which simplifies to 11⅔×D = 35 Divide both sides of the above equation by 11⅔: 11⅔×D ÷ 11⅔ = 35 ÷ 11⅔ which means D = 3 making A = 3×D = 3 × 3 = 9 B = 2×D = 2 × 3 = 6 C = F = 1⅔×D = 1⅔ × 3 = 5 E = 2⅓×D = 2⅓ × 3 = 7 and ABCDEF = 965375