Puzzle for January 21, 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.
* AB is a 2-digit number (not A×B).
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Hint #1
Add E to both sides of eq.2: C - E + E = A - C + F + E which becomes C = A - C + F + E which may be written as eq.2a) C = A - C + E + F
Hint #2
In eq.2a, replace E + F with B + C (from eq.1): C = A - C + B + C which becomes eq.1a) C = A + B
Hint #3
eq.4 may be written as: 10×A + B = C + F In the above equation, replace C with A + B (from eq.1a): 10×A + B = A + B + F Subtract A and B from each side: 10×A + B - A - B = A + B + F - A - B which simpifies to eq.4b) 9×A = F
Hint #4
Since A and F are one-digit non-negative integers, then: F ≤ 9 which means 9×A ≤ 9 (from eq.4b) making A = 0 or 1 Since A ≠ 0 (from eq.5), then: A = 1 which makes F = 9×A = 9
Hint #5
In eq.5, replace A with 1: D + E = B ÷ 1 which makes D + E = B Subtract E from each side of the above equation: D + E - E = B - E which becomes eq.5a) D = B - E
Hint #6
In eq.1a, substitute 1 for A: eq.1b) C = 1 + B
Hint #7
In eq.1, substitute 9 for F, and 1 + B for C (from eq.1b): E + 9 = B + 1 + B which becomes E + 9 = 2×B + 1 Subtract 9 from each side of the equation above: E + 9 - 9 = 2×B + 1 - 9 which becomes eq.1c) E = 2×B - 8
Hint #8
Substitute (B - E) for D (from eq.5a), 9 for F, and 1 for A in eq.3: E - (B - E) + 9 = 1 + B - E which becomes E - B + E + 9 = 1 + B - E which becomes 2×E - B + 9 = 1 + B - E In the above equation, add B and E to both sides, and subtract 1 from both sides: 2×E - B + 9 + B + E - 1 = 1 + B - E + B + E - 1 which simplifies to eq.3a) 3×E + 8 = 2×B
Hint #9
Substitute (3×E + 8) for 2×B (from eq.3a) into eq.1c: E = 3×E + 8 - 8 which becomes E = 3×E Subtract E from both sides of the equation above: E - E = 3×E - E which makes 0 = 2×E which means 0 = E
Hint #10
Substitute 0 for E in eq.3a: 3×0 + 8 = 2×B which becomes 0 + 8 = 2×B which makes 8 = 2×B Divide both sides of the above equation by 2: 8 ÷ 2 = 2×B ÷ 2 which makes 4 = B
Hint #11
Substitute 4 for B in eq.1b: C = 1 + 4 which makes C = 5
Solution
Substitute 4 for B, and 0 for E in eq.5a: D = 4 - 0 which makes D = 4 and makes ABCDEF = 145409