Puzzle for May 21, 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 positive integer.
Scratchpad
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Hint #1
Subtract the left and right sides of eq.3 from the left and right sides of eq.4, respectively: A + C – D – E – F – (B + C – D – E) = D + F – (D + E + F) which becomes A + C – D – E – F – B – C + D + E = D + F – D – E – F which simplifies to A – F – B = –E Add F to both sides of the equation above: A – F – B + F = –E + F which becomes A – B = –E + F which may be written as eq.3a) A – B = F – E
Hint #2
In eq.2, replace A – B with F – E (from eq.3a): D – E = F – E Add E to both sides of the above equation: D – E + E = F – E + E which makes D = F
Hint #3
In eq.1, replace F with D: C + D = A + D Subtract D from both sides of the above equation: C + D – D = A + D – D which makes C = A
Hint #4
eq.5 may be written as: D + F = (B + C) ÷ 2 Substitute D for F, and A for C in the equation above: D + D = (B + A) ÷ 2 which becomes 2×D = (B + A) ÷ 2 Multiply both sides by 2: 2 × 2×D = 2 × (B + A) ÷ 2 which becomes 4×D = B + A which is the same as eq.5a) 4×D = A + B
Hint #5
Substitute 4×D for A + B (from eq.5a) into eq.6: 4×D = D × F Divide both sides by D: 4×D ÷ D = (D × F) ÷ D which makes 4 = F and also makes D = F = 4
Hint #6
In eq.5a, substitute 4 for D: 4×4 = A + B which becomes eq.5b) 16 = A + B
Hint #7
Substitute A for C, and 4 for D and F in eq.3: B + A – 4 – E = 4 + E + 4 which becomes B + A – 4 – E = 8 + E Add 4 and E to both sides of the above equation: B + A – 4 – E + 4 + 4 = 8 + E + 4 + E which becomes B + A = 12 + 2×E which is the same as eq.3b) A + B = 12 + 2×E
Hint #8
Substitute 16 for A + B (from eq.5b) in eq.3b: 16 = 12 + 2×E Subtract 12 from each side of the above equation: 16 – 12 = 12 + 2×E – 12 which makes 4 = 2×E Divide both sides by 2: 4 ÷ 2 = 2×E ÷ 2 which makes 2 = E
Hint #9
In eq.4, substitute A for C, 4 for D and F, and 2 for E: A + A – 4 – 2 – 4 = 4 + 4 which becomes 2×A – 10 = 8 Add 10 to both sides of the equation above: 2×A – 10 + 10 = 8 + 10 which makes 2×A = 18 Divide both sides by 2: 2×A ÷ 2 = 18 ÷ 2 which makes A = 9 and also makes C = A = 9
Solution
Substitute 9 for A in eq.5b: 16 = 9 + B Subtract 9 from each side of the equation above: 16 – 9 = 9 + B – 9 which makes 7 = B and makes ABCDEF = 979424