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**
.