Puzzle for December 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.
Our thanks go out to Lily S (age 13) for sending us this puzzle. Thank you, Lily!
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Hint #1
In eq.4, replace A with E + F (from eq.2): B + E = E + F + F which becomes B + E = E + 2×F Subtract E from each side of the above equation: B + E – E = E + 2×F – E which makes B = 2×F
Hint #2
eq.6 may be written as: D = (A + B + C + E) ÷ 4 Multiply both sides of the above equation by 4: 4 × D = 4 × (A + B + C + E) ÷ 4 which becomes eq.6a) 4×D = A + B + C + E
Hint #3
In eq.6a, replace B + C with A + E (from eq.3): 4×D = A + A + E + E which becomes 4×D = 2×A + 2×E Divide both sides of the above equation by 2: 4×D ÷ 2 = (2×A + 2×E) ÷ 2 which becomes eq.6b) 2×D = A + E
Hint #4
In eq.5, substitute 2×F for B: D + E = 2×F + F which becomes eq.5a) D + E = 3×F Add D to both sides of eq.5a: D + E + D = 3×F + D which becomes eq.5b) 2×D + E = 3×F + D
Hint #5
Substitute A + E for 2×D (from eq.6b) in eq.5b: A + E + E = 3×F + D which becomes eq.5c) A + 2×E = 3×F + D
Hint #6
Substitute E + F for A (from eq.2) in eq.5c: E + F + 2×E = 3×F + D which becomes F + 3×E = 3×F + D Subtract 3×F from each side of the equation above: F + 3×E – 3×F = 3×F + D – 3×F which becomes eq.5d) 3×E – 2×F = D
Hint #7
Substitute 3×E – 2×F for D (from eq.5d) in eq.5a: 3×E – 2×F + E = 3×F which becomes 4×E – 2×F = 3×F Add 2×F to both sides of the equation above: 4×E – 2×F + 2×F = 3×F + 2×F which becomes 4×E = 5×F Divide both sides by 4: 4×E ÷ 4 = 5×F ÷ 4 which makes E = 1¼×F
Hint #8
Substitute 1¼×F for E in eq.2: A = 1¼×F + F which becomes A = 2¼×F
Hint #9
Substitute (1¼×F) for E in eq.5d: 3×(1¼×F) – 2×F = D which becomes 3¾×F – 2×F = D which makes 1¾×F = D
Hint #10
Substitute 2¼×F for A, 1¼×F for E, and 2×F for B in in eq.3: 2¼×F + 1¼×F = 2×F + C which becomes 3½×F = 2×F + C Subtract 2×F from each side of the equation above: 3½×F – 2×F = 2×F + C – 2×F which makes 1½×F = C
Solution
Substitute 2¼×F for A, 2×F for B, 1½×F for C, 1¾×F for D, and 1¼×F for E in in eq.1: 2¼×F + 2×F + 1½×F + 1¾×F + 1¼×F + F = 39 which simplifies to 9¾×F = 39 Divide both sides of the above equation by 9¾: 9¾×F ÷ 9¾ = 39 ÷ 9¾ which means F = 4 making A = 2¼×F = 2¼ × 4 = 9 B = 2×F = 2 × 4 = 8 C = 1½×F = 1½ × 4 = 6 D = 1¾×F = 1¾ × 4 = 7 E = 1¼×F = 1¼ × 4 = 5 and ABCDEF = 986754