Puzzle for June 25, 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
In eq.5, replace C + F with A + D + E (from eq.3): A + D + E – A = A + D – E which becomes D + E = A + D – E In the equation above, subtract D from both sides, and add E to both sides: D + E – D + E = A + D – E – D + E which simplifies to 2×E = A
Hint #2
Subtract F, D, and E from both sides of eq.3: C + F – F – D – E = A + D + E – F – D – E which becomes C – D – E = A – F Add the left and right sides of the above equation to the left and right sides of eq.2, respectively: B + E + C – D – E = A + F + A – F which becomes eq.2a) B + C – D = 2×A
Hint #3
In eq.4, add A to both sides, and subtract D from both sides: B + C – A + A – D = A – C + D + A – D which becomes B + C – D = 2×A – C In the above equation, substitute 2×A for B + C – D (from eq.2a): 2×A = 2×A – C Subtract 2×A from both sides: 2×A – 2×A = 2×A – C – 2×A which makes 0 = –C which means 0 = C
Hint #4
Substitute 0 for C in eq.2a: B + 0 – D = 2×A which becomes B – D = 2×A Add D to both sides of the equation above: B – D + D = 2×A + D which becomes eq.2b) B = 2×A + D
Hint #5
Substitute 0 for C in eq.3: 0 + F = A + D + E which becomes eq.3a) F = A + D + E
Hint #6
Substitute 2×A + D for B (from eq.2b), 0 for C, and A + D + E for F (from eq.3a) in eq.6: A + 2×A + D + E = 0 + D – E + A + D + E which becomes 3×A + D + E = 2×D + A Subtract D and A from each side of the above equation: 3×A + D + E – D – A = 2×D + A – D – A which becomes eq.6a) 2×A + E = D
Hint #7
Substitute (2×E) for A in eq.6a: 2×(2×E) + E = D which becomes 4×E + E = D which makes 5×E = D
Hint #8
Substitute (2×E) for A, and 5×E for D in eq.2b: B = 2×(2×E) + 5×E which becomes B = 4×E + 5×E which makes B = 9×E
Hint #9
Substitute 2×E for A, and 5×E for D in eq.3a: F = 2×E + 5×E + E which makes F = 8×E
Solution
Substitute 2×E for A, 9×E for B, 0 for C, 5×E for D, and 8×E for F in eq.1: 2×E + 9×E + 0 + 5×E + E + 8×E = 25 which simplifies to 25×E = 25 Divide both sides of the above equation by 25: 25×E ÷ 25 = 25 ÷ 25 which means E = 1 making A = 2×E = 2 × 1 = 2 B = 9×E = 9 × 1 = 9 D = 5×E = 5 × 1 = 5 F = 8×E = 8 × 1 = 8 and ABCDEF = 290518