I have four different things I can spread on my toast. If I want to taste them all, I need four slices of toast.

However, as we all know, the taste of jam on toast depends on what I had before. Jam after peanut butter tastes quite different from jam after vegemite. So if I have 4 different things I can spread on my toast, how many slices of toast do I need, if I want to taste every combination of X after Y ?

32 slices of toast will do the trick. There are 16 possible pairs of X after Y (Jam after jam, jam after peanut butter, jam after vegemite, jam after just margarine, and so on). Each pair has 2 slices of toast, so that’s 32 slices of toast.

But the thought of eating so much toast makes my stomach feel queasy. Is there a way I can try every combination, but with less toast? How many slices of toast must I eat, what should they have on them, and in what order must I eat them, so that I get to try all 16 combinations of X after Y?

By way of a hint, if I only have 2 things to spread, I can try all four combinations with only 5 slices of toast. I either eat

• Jam, Jam, Vegemite, Vegemite, Jam, OR
• Vegemite, Vegemite, Jam, Jam, Vegemite

depending on what I like more : sweetness or saltiness. As you can see, I get to try four combinations

• Jam after Jam,
• Jam after Vegemite,
• Vegemite after Jam, and
• Vegemite after Vegemite

but I only have to eat five pieces of toast instead of eight.

Have fun with the puzzle!