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Age 11 to 14 Challenge Level:

This one brought in lots of very neat solutions and the following one was done by Ian Green of the Coopers' Company and Coburn School.

\begin{eqnarray} b + c + d + e&=&4\\ a + c + d + e&=&5\\ a + b + d + e&=&1\\ a + b + c + e&=&2\\ a + b + c + d&=&0 \end{eqnarray}
The sum of the five equations is: $$ 4a + 4b + 4c + 4d + 4e = 12 $$ Dividing by 4: $$ a + b + c + d + e = 3 $$ We can use this equation to find the letter missing from the others (the first equation is missing a, the second b, etc.)
\begin{eqnarray} 3 - a = 4 & \Rightarrow a = -1 \\ 3 - b = 5 & \Rightarrow b = -2 \\ 3 - c = 1 & \Rightarrow c = 2 \\ 3 - d = 2 & \Rightarrow d = 1 \\ 3 - e = 0 & \Rightarrow e = 3 \end{eqnarray}

Congratulations also to M. Crompton, Lena Brown and Sophie , City of Norwich School, to Charlotte Smith, Suzanne Abbott, Christiane Eaves, Peach Pongjetanapong, Sheila Luk, Cheryl Wong , Y9, the Mount School, York to Lucy Holden, Sophie Lawler, Maria Mullen, Helen Musgrave, Y10 the Mount School, York, to Amy Mackay, Gen Cale, Sonya Rice, James Page, Jack Adcock, Geoffrey Roles, Hethersett High School, Norwich, to Thomas Wheeler King Edward VI Camp Hill School for Boys, Birmingham to Jinhyun Kim, Y7, Kingston Grammar School, to Alex Filz , Ousedale School, to Emma Dunford age 12, Wildern School, Southampton, to Michael Swarbrick Jones , Y7 Comberton Village College, Cambridgeshire, to Ipswich High School Maths Club , to Jessica Zhang , age 12, Woodbridge High School, Essex, to Claire Kruithof, Kim Murray, Sophie Palmer, Anna Beedham, Jessamy Marsden , Madras College, St Andrew's and to Kang Hong Joo .