Puzzle for April 22, 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 Abby S (age 10) for sending us this puzzle. Thank you, Abby!
Scratchpad
Help Area
Hint #1
In eq.5, replace C + E with D + F (from eq.3): A + D + F = B + D Subtract D from each side of the above equation: A + D + F – D = B + D – D which becomes eq.5a) A + F = B
Hint #2
In eq.2, replace B with A + F (from eq.5a): A + F + F = A + E Subtract A from both sides of the above equation: A + F + F – A = A + E – A which makes 2×F = E
Hint #3
In eq.4, substitute 2×F for E, and A + F for B (from eq.5a): 2×F + F = A + A + F – D which becomes 3×F = 2×A + F – D In the above equation, add D to both sides, and subtract 3×F from both sides: 3×F + D – 3×F = 2×A + F – D + D – 3×F which becomes eq.4a) D = 2×A – 2×F
Hint #4
Substitute 2×A – 2×F for D (from eq.4a), and 2×F for E in eq.3: 2×A – 2×F + F = C + 2×F which becomes 2×A – F = C + 2×F Subtract 2×F from each side of the equation above: 2×A – F – 2×F = C + 2×F – 2×F which becomes eq.3a) 2×A – 3×F = C
Hint #5
In eq.6, substitute 2×A – 3×F for C (from eq.3a), 2×A – 2×F for D (from eq.4a), A + F for B (from eq.5a), and 2×F for E: 2×A – 3×F + 2×A – 2×F – F = A + A + F + 2×F + F which becomes 4×A – 6×F = 2×A + 4×F In the equation above, subtract 2×A from both sides, and add 6×F to both sides: 4×A – 6×F – 2×A + 6×F = 2×A + 4×F – 2×A + 6×F which simplifies to 2×A = 10×F Divide both sides by 2: 2×A ÷ 2 = 10×F ÷ 2 which makes A = 5×F
Hint #6
Substitute (5×F) for A in eq.3a: 2×(5×F) – 3×F = C which becomes 10×F – 3×F = C which makes 7×F = C
Hint #7
Substitute (5×F) for A in eq.4a: D = 2×(5×F) – 2×F which becomes D = 10×F – 2×F which makes D = 8×F
Hint #8
Substitute 5×F for A in eq.5a: 5×F + F = B which makes 6×F = B
Solution
Substitute 5×F for A, 6×F for B, 7×F for C, 8×F for D, and 2×F for E in eq.1: 5×F + 6×F + 7×F + 8×F + 2×F + F = 29 which simplifies to 29×F = 29 Divide both sides of the above equation by 29: 29×F ÷ 29 = 29 ÷ 29 which means F = 1 making A = 5×F = 5 × 1 = 5 B = 6×F = 6 × 1 = 6 C = 7×F = 7 × 1 = 7 D = 8×F = 8 × 1 = 8 E = 2×F = 2 × 1 = 2 and ABCDEF = 567821