I’ve been thinking about symbol function questions today, and although it’s not easy to come up with a really hard symbol function, this one is probably harder than you’d see on the SAT. Not by much, though. Make it multiple choice and this could totally appear.

The prize this week: same as last week’s. Free access to the PWN the SAT Math Guide Beta Program. First correct answer in the comments doesn’t have to pay a measly \$5 to see what I’ve been working on tirelessly for weeks.

Let the ♠ symbol be defined such that b♠ equals the sum of the greatest two integer factors of b, for all integers b. If &lt; 100, what is the greatest possible value of x♠?

Good luck! I’ll post the solution Monday.

UPDATE: Nice work, Anu! Hope you enjoy the Math Guide Beta. Solution posted below the cut.

This was a bit of a tricky question. To get a handle on it, just start listing factors of the highest possible values of x:

Factors of 99: 99, 33, 11, 9, 3, 1. So 99♠ = 99 + 33 = 132.

Factors of 98: 98, 49, 14, 7, 2, 1. So 98♠ = 98 + 49 = 147.

Factors of 97: 97, 1 (97 is prime). So 97♠ = 97 + 1 = 98.

Factors of 96: 96, 48, 32, 24, 16, 12, 8, 6, 4, 3, 2, 1. So 96♠ = 96 + 48 = 144.

Are you seeing a pattern yet? x♠ will always be bigger for an even x than for an adjacent odd x. So the highest even x allowed will give us our greatest value. That’s 98♠, or 147.