Puzzle for July 24, 2020 ( )
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 C with B + E (from eq.2): B + E – E = B – D + F which becomes B = B – D + F In the equation above, subtract B from both sides, and add D to both sides: B – B + D = B – D + F – B + D which simplifies to D = F
Hint #2
Subtract B from each side of eq.2: B + E – B = C – B which becomes E = C – B In eq.4, replace E with C – B: A – B = B + C + C – B which becomes A – B = 2×C Add B to both sides of the equation above: A – B + B = 2×C + B which becomes eq.4a) A = 2×C + B
Hint #3
In eq.4a, substitute (B + E) for C (from eq.2): A = 2×(B + E) + B which is the same as A = 2×B + 2×E + B which becomes eq.4b) A = 3×B + 2×E
Hint #4
Substitute D for F in eq.5: D + E = A – C – E + D In the equation above, subtract D from both sides, and add C and E to both sides: D + E – D + C + E = A – C – E + D – D + C + E which simplifies to eq.5a) 2×E + C = A
Hint #5
Substitute B + E for C (from eq.2) in eq.5a: 2×E + B + E = A which makes eq.5b) B + 3×E = A
Hint #6
Substitute 3×B + 2×E for A in (from eq.4b) in eq.5b: B + 3×E = 3×B + 2×E Subtract B and 2×E from both sides of the above equation: B + 3×E – B – 2×E = 3×B + 2×E – B – 2×E which makes E = 2×B
Hint #7
Substitute 2×B for E in eq.2: B + 2×B = C which makes 3×B = C
Hint #8
Substitute 3×B for C, and 2×B for E in eq.6: 3×B + 2×B – F = B – 3×B which becomes 5×B – F = –2×B Add F and 2×B to both sides of the equation above: 5×B – F + F + 2×B = –2×B + F + 2×B which makes 7×B = F and also makes D = F = 7×B
Hint #9
Substitute (2×B) for E in eq.4b: A = 3×B + 2×(2×B) which is equivalent to A = 3×B + 4×B which becomes A = 7×B
Solution
Substitute 7×B for A and D and F, 3×B for C, and 2×B for E in eq.1: 7×B + B + 3×B + 7×B + 2×B + 7×B = 27 which simplifies to 27×B = 27 Divide both sides of the equation above by 27: 27×B ÷ 27 = 27 ÷ 27 which means B = 1 making A = D = F = 7×B = 7 × 1 = 7 C = 3×B = 3 × 1 = 3 E = 2×B = 2 × 1 = 2 and ABCDEF = 713727