Puzzle for November 29, 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.
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Hint #1
In eq.6, replace E + F with A + C (from eq.3): D + A + C = A + B + C Subtract A and C from each side of the equation above: D + A + C - A - C = A + B + C - A - C which becomes D = B
Hint #2
In eq.4, replace D with B: C - B = B + B which becomes C - B = 2×B Add B to both sides of the above equation: C - B + B = 2×B + B which makes C = 3×B
Hint #3
In eq.5, substitute B for D: A - B = B + B + F which becomes A - B = 2×B + F Add B to both sides of the equation above: A - B + B = 2×B + F + B which becomes eq.5a) A = 3×B + F
Hint #4
In eq.5a, substitute C + E for A (from eq.2), and C for 3×B: C + E = C + F Subtract C from each side of the above equation: C + E - C = C + F - C which makes E = F
Hint #5
Substitute F for E, 3×B + F for A (from eq.5a), and 3×B for C in eq.2: F + F = 3×B + F + 3×B which becomes 2×F = 6×B + F Subtract F from both sides of the above equation: 2×F - F = 6×B + F - F which makes F = 6×B and also makes E = F = 6×B
Hint #6
Substitute 6×B for F in eq.5a: A = 3×B + 6×B which makes A = 9×B
Solution
Substitute 9×B for A, 3×B for C, B for D, and 6×B for E and F in eq.1: 9×B + B + 3×B + B + 6×B + 6×B = 26 which simplifies to 26×B = 26 Divide both sides of the above equation by 26: 26×B ÷ 26 = 26 ÷ 26 which means B = 1 making A = 9×B = 9 × 1 = 9 C = 3×B = 3 × 1 = 3 D = B = 1 E = F = 6×B = 6 × 1 = 6 and ABCDEF = 913166