One of the incentives to join the army is their repayment program:

“The Loan Repayment Program (LRP) is a special enlistment incentive offered to qualified applicants at the time of enlistment. Under the LRP, the Army will repay one-third of your loan for each year of full-time duty served.” (see http://www.goarmy.com/benefits/education-benefits/money-for-college.html)

My understanding is that this 1/3 that they pay off each year is taxable, but the full amount (subject to a limit) is paid off in three years.  On the other hand the public service program takes 10 years, but is not taxable at the end.  How does one go about calculating which makes more sense?

Sorry it’s taken me a while to get back to you.  I was doing some research on the Army LRP and am having a hard time accessing the authorizing legislation.  I have dug up the regulation, read Chapter 4 here: http://www.usarec.army.mil/im/formpub/REC_PUBS/R621_1.pdf

Here are some things to think about:
-make sure you meet the LRP eligibility criteria (declined GI Bill in writing, LRP guaranteed in writing in your enlistment contract, etc.)
-bear in mind the $65,000 lifetime cap on benefits
-note than ONLY principle is repaid, NOT interest (which will accrue and fast)
-and as you note, the benefits are taxable as income to you (I’m not a tax expert and your tax obligation will depend on lots of other factors, but you could pay as much as 1/3 of the benefits back in taxes.  Could be less, I just can’t say).