Puzzle for January 25, 2021 ( )
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.5, replace E with A + D (from eq.3): A + D – D = A + C which becomes A = A + C Subtract A from both sides of the equation above: A – A = A + C – A which makes 0 = C
Hint #2
In eq.2, replace C with 0: 0 + D = A which makes D = A
Hint #3
In eq.3, substitute A for D: A + A = E which makes eq.3a) 2×A = E
Hint #4
In eq.6, substitute 0 for C: B – 0 – F = 0 – B which becomes B – F = –B Add F and B to both sides of the above equation: B – F + F + B = –B + F + B which makes 2×B = F
Hint #5
Substitute 2×B for F in eq.4: 2×B – E = E – B Add E and B to both sides of the equation above: 2×B – E + E + B = E – B + E + B which becomes 3×B = 2×E Divide both sides by 2: 3×B ÷ 2 = 2×E ÷ 2 which makes 1½×B = E
Hint #6
Substitute 1½×B for E in eq.3a: 2×A = 1½×B Divide each side by 2: 2×A ÷ 2 = 1½×B ÷ 2 which makes A = ¾×B and also makes D = A = ¾×B
Solution
Substitute ¾×B for A and D, 0 for C, 1½×B for E, and 2×B for F in eq.1: ¾×B + B + 0 + ¾×B + 1½×B + 2×B = 24 which simplifies to 6×B = 24 Divide both sides of the equation above by 6: 6×B ÷ 6 = 24 ÷ 6 which means B = 4 making A = D = ¾×B = ¾ × 4 = 3 E = 1½×B = 1½ × 4 = 6 F = 2×B = 2 × 4 = 8 and ABCDEF = 340368