Puzzle for May 5, 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
Add B and C to both sides of eq.4: D – B + B + C = B – C + E + B + C which becomes D + C = 2×B + E which may be written as eq.4a) C + D = 2×B + E In eq.6, replace C + D with 2×B + E (from eq.4a): A = (2×B + E – E) ÷ B which becomes A = (2×B) ÷ B which makes A = 2
Hint #2
Add A and B to both sides of eq.5: B – A + A + B = D + F – B – E + A + B which becomes 2×B = D + F – E + A which may be written as eq.5a) 2×B = A + D + F – E In eq.5a, replace A + D with C + E (from eq.3): 2×B = C + E + F – E which becomes eq.5b) 2×B = C + F
Hint #3
In eq.4a, substitute C + F for 2×B (from eq.5b): C + D = C + F + E Subtract C from each side of the equation above: C + D – C = C + F + E – C which becomes eq.4b) D = F + E
Hint #4
Substitute D + E for F (from eq.2) into eq.4b: D = D + E + E which becomes D = D + 2×E Subtract D from each side of the above equation: D – D = D + 2×E – D which makes 0 = 2×E which means 0 = E
Hint #5
Substitute 0 for E in eq.2: F = D + 0 which makes F = D
Hint #6
Substitute 2 for A, D for F, and 0 for E in eq.5a: 2×B = 2 + D + D – 0 which becomes 2×B = 2 + 2×D Divide both sides of the above equation by 2: 2×B ÷ 2 = (2 + 2×D) ÷ 2 which makes eq.5c) B = 1 + D
Hint #7
Substitute (1 + D) for B (from eq.5c), and D for F in eq.5b: 2×(1 + D) = C + D which becomes 2 + 2×D = C + D Subtract D from each side of the equation above: 2 + 2×D – D = C + D – D which makes eq.5d) 2 + D = C
Solution
Substitute 2 for A, 1 + D for B (from eq.5c), 2 + D for C (from eq.5d), 0 for E, and D for F in eq.1: 2 + 1 + D + 2 + D + D + 0 + D = 33 which simplifies to 5 + 4×D = 33 Subtract 5 from both sides of the equation above: 5 + 4×D – 5 = 33 – 5 which makes 4×D = 28 Divide both sides by 4: 4×D ÷ 4 = 28 ÷ 4 which means D = 7 making B = 1 + D = 1 + 7 = 8 (from eq.5c) C = 2 + D = 2 + 7 = 9 (from eq.5d) F = D = 7 and ABCDEF = 289707