Puzzle for February 12, 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.5, replace A + B with F (from eq.2): D + F – B = F + C In the above equation, subtract F from both sides, and add B to both sides: D + F – B – F + B = F + C – F + B which becomes eq.5a) D = C + B
Hint #2
In eq.3, replace D with C + B (from eq.5a): B + E = C + B – E In the equation above, subtract B from both sides, and add E to both sides: B + E – B + E = C + B – E – B + E which makes 2×E = C
Hint #3
In eq.4, substitute 2×E for C: A + E = B – 2×E Add 2×E to both sides of the equation above: A + E + 2×E = B – 2×E + 2×E which becomes eq.4a) A + 3×E = B
Hint #4
Substitute A + 3×E for B (from eq.4a) in eq.2: F = A + A + 3×E which becomes eq.2a) F = 2×A + 3×E
Hint #5
Substitute A + 3×E for B (from eq.4a) in eq.3: A + 3×E + E = D – E which becomes A + 4×E = D – E Add E to both sides of the above equation: A + 4×E + E = D – E + E which becomes eq.3a) A + 5×E = D
Hint #6
Substitute A + 3×E for B (from eq.4a), 2×E for C, A + 5×E for D (from eq.3a), and 2×A + 3×E for F (from eq.2a) in eq.1: A + A + 3×E + 2×E + A + 5×E + E + 2×A + 3×E = 24 which becomes 5×A + 14×E = 24 Subtract 14×E from each side of the equation above: 5×A + 14×E – 14×E = 24 – 14×E which becomes 5×A = 24 – 14×E Divide both sides by 5: 5×A ÷ 5 = (24 – 14×E) ÷ 5 which becomes eq.1a) A = (24 – 14×E) ÷ 5
Hint #7
To make eq.1a true, check several possible values for E and A: If E = 0, then A = (24 – 14×0) ÷ 5 = (24 – 0) ÷ 5 = 24 ÷ 5 = 4⅘ If E = 1, then A = (24 – 14×1) ÷ 5 = (24 – 14) ÷ 5 = 10 ÷ 5 = 2 If E = 2, then A = (24 – 14×2) ÷ 5 = (24 – 28) ÷ 5 = –4 ÷ 5 = –⅘ If E > 2, then A < –⅘ Since A must be a one-digit non-negative integer, then A = 2 which means E = 1 and makes C = 2×E = 2×1 = 2
Hint #8
Substitute 2 for A, and 1 for E in eq.4a: 2 + 3×1 = B which becomes 2 + 3 = B which means 5 = B
Hint #9
Substitute 2 for A, and 1 for E in eq.3a: 2 + 5×1 = D which becomes 2 + 5 = D which means 7 = D
Solution
Substitute 2 for A, and 5 for B in eq.2: F = 2 + 5 which means F = 7 and ABCDEF = 252717