Puzzle for November 19, 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.
* DE and CD are 2-digit numbers (not D×E or C×D).
Scratchpad
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Hint #1
eq.6 may be written as: C = (D + E + F) ÷ 3 Multiply both sides of the above equation by 3: 3 × C = 3 × (D + E + F) ÷ 3 which becomes eq.6a) 3×C = D + E + F
Hint #2
In eq.6a, replace D + E with C + F (from eq.3): 3×C = C + F + F which becomes 3×C = C + 2×F Subtract C from each side of the equation above: 3×C – C = C + 2×F – C which becomes 2×C = 2×F Divide both sides by 2: 2×C ÷ 2 = 2×F ÷ 2 which makes C = F
Hint #3
eq.5 may be written as: 10×D + E = 10×C + D – F In the above equation, replace F with C: 10×D + E = 10×C + D – C which becomes 10×D + E = 9×C + D Subtract 10×D from each side: 10×D + E – 10×D = 9×C + D – 10×D which becomes E = 9×F – 9×D which may be written as eq.5a) E = 9×(F – D)
Hint #4
In eq.4, replace A + B with F (from eq.2), and C with F: F + D = F + F – B which becomes F + D = 2×F – B In the above equation, add B to both sides, and subtract F and D from both sides: F + D + B – F – D = 2×F – B + B – F – D which simplifies to eq.4a) B = F – D
Hint #5
Substitute B for F – D (from eq.4a) into eq.5a: E = 9×(B) which is the same as E = 9×B
Hint #6
Substitute A + B for F (from eq.2) in eq.4a: B = A + B – D In the equation above, subtract B from both sides, and add D to both sides: B – B + D = A + B – D – B + D which simplifies to D = A
Hint #7
Substitute A for D, 9×B for E, and F for C in eq.3: A + 9×B = F + F which becomes eq.3a) A + 9×B = 2×F
Hint #8
Substitute (A + B) for F (from eq.2) into eq.3a: A + 9×B = 2×(A + B) which becomes A + 9×B = 2×A + 2×B Subtract A and 2×B from both sides of the equation above: A + 9×B – A – 2×B = 2×A + 2×B – A – 2×B which makes 7×B = A and also makes D = A = 7×B
Hint #9
Substitute 7×B for A in eq.2: F = 7×B + B which makes F = 8×B and also makes C = F = 8×B
Solution
Substitute 7×B for A and D, 8×B for C and F, and 9×B for E in eq.1: 7×B + B + 8×B + 7×B + 9×B + 8×B = 40 which simplifies to 40×B = 40 Divide both sides of the above equation by 40: 40×B ÷ 40 = 40 ÷ 40 which means B = 1 making A = D = 7×B = 7 × 1 = 7 C = F = 8×B = 8 × 1 = 8 E = 9×B = 9 × 1 = 9 and ABCDEF = 718798