Puzzle for June 23, 2020 ( )
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 C and E to both sides of eq.4: A + C + E = D – C – E + C + E which becomes A + C + E = D In the equation above, replace A + C with E + F (from eq.3): E + F + E = D which becomes eq.4a) 2×E + F = D
Hint #2
In eq.4a, replace F with D + E (from eq.2): 2×E + D + E = D which becomes 3×E + D = D Subtract D from each side of the above equation: 3×E + D – D = D – D which makes 3×E = 0 which means E = 0
Hint #3
In eq.2, substitute 0 for E: D + 0 = F which makes D = F
Hint #4
Substitute D for F in eq.6: C + D – A = D – C Subtract D from both sides of the equation above: C + D – A – D = D – C – D which becomes C – A = –C Add A and C to both sides: C – A + A + C = –C + A + C which becomes 2×C = A
Hint #5
Substitute 2×C for A, and 0 for E in eq.3: 2×C + C = 0 + F which makes 3×C = F and also makes D = F = 3×C
Hint #6
Substitute 3×C for F in eq.5: 3×C – B + C = B – (3×C – B) which becomes 4×C – B = B – 3×C + B which becomes 4×C – B = 2×B – 3×C Add B and 3×C to both sides of the equation above: 4×C – B + B + 3×C = 2×B – 3×C + B + 3×C which makes 7×C = 3×B Divide both sides by 3: 7×C ÷ 3 = 3×B ÷ 3 which makes 2⅓×C = B
Solution
Substitute 2×C for A, 2⅓×C for B, 3×C for D and F, and 0 for E in eq.1: 2×C + 2⅓×C + C + 3×C + 0 + 3×C = 34 which simplifies to 11⅓×C = 34 Divide both sides of the above equation by 11⅓: 11⅓×C ÷ 11⅓ = 34 ÷ 11⅓ which means C = 3 making A = 2×C = 2 × 3 = 6 B = 2⅓×C = 2⅓ × 3 = 7 D = F = 3×C = 3 × 3 = 9 and ABCDEF = 673909