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 ofb, for all integersb. Ifx< 100, what is the greatest possible value ofx♠?

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.

## Comments (7)

147

There is no reason the value of x-spade has to be an integer, is there? I’ll go for 148.5

I think the answer is 51.

Is it.. 80?

147

Well done. Welcome to the Beta. (You should see it if you log into docs.google.com)

147