Puzzle for June 3, 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 B and F to both sides of eq.2: B - F + B + F = A - B + B + F which becomes eq.2a) 2×B = A + F Add B and F to both sides of eq.5: E + F - B + B + F = B + C - F + B + F which becomes eq.5a) E + 2×F = 2×B + C
Hint #2
In eq.5a, replace 2×B with A + F (from eq.2a): E + 2×F = A + F + C Subtract F from each side of the equation above: E + 2×F - F = A + F + C - F which becomes eq.5b) E + F = A + C
Hint #3
In eq.6, replace E + F with A + C (from eq.5b): A + B + C = D + A + C - C which becomes A + B + C = D + A Subtract A from both sides of the equation above: A + B + C - A = D + A - A which becomes eq.6a) B + C = D
Hint #4
In eq.3, substitute D for B + C (from eq.6a): D + F = D Subtract D from each side of the equation above: D + F - D = D - D which makes F = 0
Hint #5
Substitute 0 for F in eq.2a: 2×B = A + 0 which makes 2×B = A
Hint #6
Substitute 2×B for A in eq.4: C - 2×B = 2×B - D + E Add 2×B to both sides of the above equation: C - 2×B + 2×B = 2×B - D + E + 2×B which becomes eq.4a) C = 4×B - D + E
Hint #7
Substitute 0 for F, and 2×B for A in eq.5b: E + 0 = 2×B + C which becomes eq.5c) E = 2×B + C
Hint #8
Substitute 2×B + C for E (from eq.5c) in eq.4a: C = 4×B - D + 2×B + C which becomes C = 6×B - D + C In the equation above, subtract C from both sides, and add D to both sides: C - C + D = 6×B - D + C - C + D which makes D = 6×B
Hint #9
Substitute 6×B for D in eq.6a: B + C = 6×B Subtract B from each side of the above equation: B + C - B = 6×B - B which makes C = 5×B
Hint #10
Substitute 5×B for C in eq.5c: E = 2×B + 5×B which makes E = 7×B
Solution
Substitute 2×B for A, 5×B for C, 6×B for D, 7×B for E, and 0 for F in eq.1: 2×B + B + 5×B + 6×B + 7×B + 0 = 21 which simplifies to 21×B = 21 Divide both sides of the above equation by 21: 21×B ÷ 21 = 21 ÷ 21 which means B = 1 making A = 2×B = 2 × 1 = 2 C = 5×B = 5 × 1 = 5 D = 6×B = 6 × 1 = 6 E = 7×B = 7 × 1 = 7 and ABCDEF = 215670