Puzzle for April 15, 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.
* EF is a 2-digit number (not E×F).
Scratchpad
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Hint #1
Add A and F to both sides of eq.3: B + C - A + A + F = A + E - F + A + F which becomes eq.3a) B + C + F = 2×A + E
Hint #2
eq.6 may be written as: A = (B + C + F) ÷ 3 Multiply both sides of the above equation by 3: 3 × A = 3 × (B + C + F) ÷ 3 which becomes eq.6a) 3×A = B + C + F
Hint #3
In eq.3a, replace B + C + F with 3×A (from eq.6a): 3×A = 2×A + E Subtract 2×A from each side of the equation above: 3×A - 2×A = 2×A + E - 2×A which makes A = E
Hint #4
In eq.2, replace E with A: A + C = A + F Subtract A from each side of the above equation: A + C - F = A + F - F which makes C = F
Hint #5
In eq.6a, substitute C for F: 3×A = B + C + C which becomes eq.6b) 3×A = B + 2×C
Hint #6
eq.5 may be written as: eq.5a) 10×E + F = A + B + D
Hint #7
Substitute A for E, and C for F in eq.5a: 10×A + C = A + B + D Subtract A from both sides of the equation above: 10×A + C - A = A + B + D - A which becomes 9×A + C = B + D which may be written as eq.5b) 3×(3×A) + C = B + D
Hint #8
Substitute B + 2×C for 3×A (from eq.6b) into eq.5b: 3×(B + 2×C) + C = B + D which becomes 3×B + 6×C + C = B + D which becomes 3×B + 7×C = B + D Subtract B from both sides of the above equation: 3×B + 7×C - B = B + D - B which becomes eq.5c) 2×B + 7×C = D
Hint #9
Substitute C for F in eq.4: D + C - B = B + C - C which becomes D + C - B = B Add B to both sides of the equation above: D + C - B + B = B + B which becomes eq.4a) D + C = 2×B
Hint #10
Substitute D + C for 2×B (from eq.4a) into eq.5c: D + C + 7×C = D which becomes D + 8×C = D Subtract D from each side of the above equation: D + 8×C - D = D - D which makes 8×C = 0 which means C = 0 and also means C = F = 0
Hint #11
Substitute 0 for C in eq.6b: 3×A = B + 2×0 which becomes 3×A = B + 0 which makes 3×A = B
Hint #12
Substitute 0 for C, and (3×A) for B in eq.4a: D + 0 = 2×(3×A) which makes D = 6×A
Solution
Substitute 3×A for B, 0 for C and F, 6×A for D, and A for E in eq.1: A + 3×A + 0 + 6×A + A + 0 = 11 which simplifies to 11×A = 11 Divide both sides of the above equation by 11: 11×A ÷ 11 = 11 ÷ 11 which means A = 1 making B = 3×A = 3 × 1 = 3 D = 6×A = 6 × 1 = 6 E = A = 1 and ABCDEF = 130610