Puzzle for May 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
Subtract D from each side of eq.2: C – D = A + D + E – D which becomes C – D = A + E In eq.6, replace A + E with C – D: B + C – E = C – D In the above equation, subtract C from both sides, and add E and D to both sides: B + C – E – C + E + D = C – D – C + E + D which simplifies to eq.6a) B + D = E
Hint #2
In eq.4, replace E with B + D (from eq.6a): B + F = D + B + D which becomes B + F = 2×D + B Subtract B from each side of the equation above: B + F – B = 2×D + B – B which makes eq.4a) F = 2×D
Hint #3
In eq.3, substitute 2×D for F (from eq.4a): E + 2×D = A + C Subtract A from each side of the above equation: E + 2×D – A = A + C – A which becomes eq.3a) E + 2×D – A = C
Hint #4
Substitute E + 2×D – A for C (from eq.3a) in eq.2: E + 2×D – A = A + D + E In the above equation, subtract E and D from both sides, and add A to both sides: E + 2×D – A – E – D + A = A + D + E – E – D + A which simplifies to D = 2×A
Hint #5
Substitute (2×A) for D in eq.4a: F = 2×(2×A) which makes F = 4×A
Hint #6
Substitute 2×A for D in eq.5: A + 2×A = B – A which becomes 3×A = B – A Add A to both sides of the above equation: 3×A + A = B – A + A which makes 4×A = B
Hint #7
Substitute 4×A for B, and 2×A for D in eq.6a: 4×A + 2×A = E which makes 6×A = E
Hint #8
Substitute 2×A for D, and 6×A for E in eq.2: C = A + 2×A + 6×A which makes C = 9×A
Solution
Substitute 4×A for B and F, 9×A for C, 2×A for D, and 6×A for E in eq.1: A + 4×A + 9×A + 2×A + 6×A + 4×A = 26 which simplifies to 26×A = 26 Divide both sides by 26: 26×A ÷ 26 = 26 ÷ 26 which means A = 1 making B = F = 4×A = 4×1 = 4 C = 9×A = 9×1 = 9 D = 2×A = 2×1 = 2 E = 6×A = 6×1 = 6 and ABCDEF = 149264