Puzzle for December 28, 2022 ( )
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.3, replace D with E + F (from eq.2): C = E + F + F which becomes eq.3a) C = E + 2×F
Hint #2
In eq.4, replace C with E + 2×F (from eq.3a): B = E + 2×F + F which becomes eq.4a) B = E + 3×F
Hint #3
In eq.5, replace B with E + 3×F (from eq.4a): A = E + 3×F + F which becomes eq.5a) A = E + 4×F
Hint #4
In eq.6, replace A with E + 4×F (from eq.5a): E = E + 4×F + F - E E = 5×F
Hint #5
In eq.2, substitute 5×F for E: D = 5×F + F which makes D = 6×F
Hint #6
In eq.3a, substitute 5×F for E: C = 5×F + 2×F which makes C = 7×F
Hint #7
In eq.4a, substitute 5×F for E: B = 5×F + 3×F which makes B = 8×F
Hint #8
In eq.5a, substitute 5×F for E: A = 5×F + 4×F which makes A = 9×F
Solution
Substitute 9×F for A, 8×F for B, 7×F for C, 6×F for D, and 5×F for E in eq.1: 9×E + 8×E + 7×E + 6×E + 5×E + E = 36 which simplifies to 36×F = 36 Divide both sides of the above equation by 36: 36×F ÷ 36 = 36 ÷ 36 which means F = 1 making A = 9×F = 9 × 1 = 9 B = 8×F = 8 × 1 = 8 C = 7×F = 7 × 1 = 7 D = 6×F = 6 × 1 = 6 E = 5×F = 5 × 1 = 5 and ABCDEF = 987651