Puzzle for April 8, 2023 ( )
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 D + E with A + B (from eq.4): B + C = A + B + F Subtract B from both sides of the above equation: B + C - B = A + B + F - B which becomes eq.5a) C = A + F
Hint #2
In eq.2, replace C with A + F (from eq.5a): A + F = B + F Subtract F from each side of the above equation: A + F - F = B + F - F which makes A = B
Hint #3
eq.1 may be written as: A + B + D + C + E + F = 32 In the above equation, substitute C + E + F for A + B + D (from eq.6): C + E + F + C + E + F = 32 which may be written as: 2×(C + E + F) = 32 Divide both sides by 2: 2×(C + E + F) ÷ 2 = 32 ÷ 2 which becomes eq.1a) C + E + F = 16
Hint #4
Substitute C for A + F (from eq.5a) in eq.3: E - F = C Add F to both sides of the equation above: E - F + F = C + F which becomes eq.3a) E = C + F
Hint #5
eq.1a may be written as: E + C + F = 16 Substitute E for C + F (from eq.3a) into the above equation: E + E = 16 which becomes 2×E = 16 Divide both sides by 2: 2×E ÷ 2 = 16 ÷ 2 which makes E = 8
Hint #6
Substitute 8 for E in eq.3a: 8 = C + F Subtract F from each side of the above equation: 8 - F = C + F - F which becomes eq.3b) 8 - F = C
Hint #7
Substitute 8 - F for C (from eq.3b) in eq.2: 8 - F = B + F Subtract F from each side of the equation above: 8 - F - F = B + F - F which makes 8 - 2×F = B and also makes eq.2a) 8 - 2×F = B = A
Hint #8
Substitute 8 for E, and 8 - 2×F for A and B (from eq.2a) in eq.4: D + 8 = 8 - 2×F + 8 - 2×F which becomes D + 8 = 16 - 4×F Subtract 8 from both sides of the equation above: D + 8 - 8 = 16 - 4×F - 8 which becomes eq.4a) D = 8 - 4×F
Hint #9
Substitute 8 - 2×F for A and B (from eq.2a), 8 - F for C (from eq.3b), 8 - 4×F for D (from eq.4a), and 8 for E in eq.1: 8 - 2×F + 8 - 2×F + 8 - F + 8 - 4×F + 8 + F = 32 which simplifies to 40 - 8×F = 32 In the above equation, add 8×F to both sides, and subtract 32 from both sides: 40 - 8×F + 8×F - 32 = 32 + 8×F - 32 which makes 8 = 8×F Divide both sides by 8: 8 ÷ 8 = 8×F ÷ 8 which makes 1 = F
Solution
Since F = 1, then: A = B = 8 - 2×F = 8 - 2×1 = 8 - 2 = 6 (from eq.2a) C = 8 - F = 8 - 1 = 7 (from eq.3b) D = 8 - 4×F = 8 - 4×1 = 8 - 4 = 4 (from eq.4a) and ABCDEF = 667481