Good Math

Puzzling Symmetry

Let’s play a game that I’m calling crypto-addition. I’ve taken the letters a through k and I’ve assigned each one to one of the numbers between -5 and +5, and I want you to figure out which letter represents which number. The only clues come from Table 3, ​A crypto-addition table​, where I’ve shown you the sums of different letters whenever that sum is between -5 and +5.

Table 3. A crypto-addition table

The letters a through k represent the numbers between -5 and +5. Entries in the table represent the sums of the row and column labels.

a	b	c	d	e	f	g	h	i	j	k
a:			b	a	h		f		d	k
b:			f	b	d		k	a	c	h
c:	b	f	g	c	i	k	j	d		e	h
d:	a	b	c	d	e	f	g	h	i	j	k
e:	h	d		e		c		j			g
f:			k	f	c		h	b	g	d	a
g:	f	k	j	e		h	e	c		i	d
h:		a	d	h	j	b	c	k	e	g	f
i:	d	c		i		g		e			j
j:	k	1	e	j		d	i	g			c
k:			h	k	g	a	d	f	j	c	b

Using the crypto-addition table, what ...