Puzzle for June 2, 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
Add F to both sides of eq.2: F - D + F = A + B - F + F which becomes eq.2a) 2×F - D = A + B Add A and F to both sides of eq.3: B - F + A + F = C + F - A + A + F which becomes B + A = C + 2×F which may be written as eq.3a) A + B = C + 2×F
Hint #2
In eq.3a, replace A + B with 2×F - D (from eq.2a): 2×F - D = 2×F + C In the above equation, subtract 2×F from both sides, and add D to both sides: 2×F - D - 2×F + D = 2×F + C - 2×F + D which makes 0 = C + D Since C and D are non-negative, the above equation makes: C = 0 and D = 0
Hint #3
In eq.3a, replace C with 0: A + B = 0 + 2×F which becomes eq.3b) A + B = 2×F
Hint #4
In eq.5, substitute 0 for D and C, and 2×F for A + B (from eq.3b): 0 + E - A = 2×F + 0 - F which becomes eq.5a) E - A = F
Hint #5
In eq.4, substitute 0 for C and D: 0 + E - B = B - 0 + F which becomes E - B = B + F Add B to both sides of the equation above: E - B + B = B + F + B which becomes eq.4a) E = 2×B + F
Hint #6
Substitute 2×B + F for E (from eq.4a) in eq.5a: 2×B + F - A = F In the above equation, subtract F from both sides, and add A to both sides: 2×B + F - A - F + A = F - F + A which makes 2×B = A
Hint #7
Substitute 2×B for A in eq.3b: 2×B + B = 2×F which makes 3×B = 2×F Divide both sides of the above equation by 2: 3×B ÷ 2 = 2×F ÷ 2 which makes 1½×B = F
Hint #8
Substitute 1½×B for F in eq.4a: E = 2×B + 1½×B which makes E = 3½×B
Solution
Substitute 2×B for A, 0 for C and D, 3½×B for E, and 1½×B for F in eq.1: 2×B + B + 0 + 0 + 3½×B + 1½×B = 16 which simplifies to 8×B = 16 Divide both sides of the above equation by 8: 8×B ÷ 8 = 16 ÷ 8 which means B = 2 making A = 2×B = 2 × 2 = 4 E = 3½×B = 3½ × 2 = 7 F = 1½×B = 1½ × 2 = 3 and ABCDEF = 420073