Puzzle for January 5, 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 C + D with E + F (from eq.3): B + E = E + F – B In the above equation, subtract E from both sides, and add B to both sides: B + E – E + B = E + F – B – E + B which becomes 2×B = F
Hint #2
eq.2 may be written as: C = A + F + B In the above equation, replace A + F with D – A (from eq.4): C = D – A + B Add A to both sides: C + A = D – A + B + A which becomes C + A = D + B which is the same as eq.2a) A + C = B + D
Hint #3
eq.6 may be re-written as: D + E – C = A + C + B + F In the equation above, substitute B + D for A + C (from eq.2a), and 2×B for F: D + E – C = B + D + B + 2×B which becomes D + E – C = D + 4×B Subtract D from both sides, and add C to both sides: D + E – C – D + C = D + 4×B – D + C which becomes eq.6a) E = 4×B + C
Hint #4
Substitute 4×B + C for E (from eq.6a), and 2×B for F in eq.3: 4×B + C + 2×B = C + D which becomes 6×B + C = C + D Subtract C from each side of the equation above: 6×B + C – C = C + D – C which makes 6×B = D
Hint #5
Substitute 2×B for F, and 6×B for D in eq.4: A + 2×B = 6×B – A In the above equation, subtract 2×B from both sides, and add A to both sides: A + 2×B – 2×B + A = 6×B – A – 2×B + A which becomes 2×A = 4×B Divide both sides by 2: 2×A ÷ 2 = 4×B ÷ 2 which makes A = 2×B
Hint #6
Substitute 2×B for A, and 6×B for D in eq.2a: 2×B + C = B + 6×B which becomes 2×B + C = 7×B Subtract 2×B from each side of the equation above: 2×B + C – 2×B = 7×B – 2×B which makes C = 5×B
Hint #7
Substitute 5×B for C in eq.6a: E = 4×B + 5×B which makes E = 9×B
Solution
Substitute 2×B for A and F, 5×B for C, 6×B for D, and 9×B for E in eq.1: 2×B + B + 5×B + 6×B + 9×B + 2×B = 25 which simplifies to 25×B = 25 Divide both sides of the above equation by 25: 25×B ÷ 25 = 25 ÷ 25 which means B = 1 making A = F = 2×B = 2 × 1 = 2 C = 5×B = 5 × 1 = 5 D = 6×B = 6 × 1 = 6 E = 9×B = 9 × 1 = 9 and ABCDEF = 215692