Let me show you a mathematical approach. Common sense dictates that the minute hand moves at a faster rate of 5.5 degrees a minute (because the hour hand moves 0.5 degrees a min and the minute hand moves 6 degrees a minute). We start at 12 midnight. The hands are together. For subsequent 90 degree angles to occur, the minute hand must "overtake" the hour hand by 90 degrees, then 270 degrees, then 360 + 90 degrees, then 360 + 270 degrees, then 360 + 360 +90 degrees.. and so on. This can be re-expressed as: (1)90, 3(90), 5(90), 7(90), 9(90), 11(90)... n(90). The number of minutes this takes to happen can be expressed as (1)90/5.5, 3(90)/5.5, 5(90)/5.5, 7(90)/5.5, 9(90)/5.5, 11(90)/5.5... n(90)/5.5. In one day, there are 24 hr * 60 mins = 1440mins To find the maximum value of n, n(90)/5.5 = 1440 n = 88 but as seen from above, n must be an odd number (by pattern recognition and logic) hence n must be the next smallest odd number (87) counting 1,3,5,7,9,11......87, we see that the number of terms = (87-1)/2 +1 = 44. In other words, the minute hand "overtakes" the hour hand on 44 occasions in 24 hours in order to give a 90 degree angle. Therefore the answer to your question is 44.

i agree with right_ans. although i got lost in his explanation, though i'm sure it is correct. i found another answer here: http://brainteaserbible.com/

Each hour has 2 occurrences of 90 degrees. In 12 hrs, it overtakes 24 times. BUT hrs 2 to 3 has only 1 NOT 2. Also hr 8 to 9 has only 1. So subtract 2 from 24. You get 22. In half a day (12 hrs) you get 22 times. Therefore in 1 day ie 24 hrs, it cross 22 * 2 = 44 times.

Relative speed is 5.5 degree/min. Time is 24*60 mins. Total distance is 5.5*24*60 degrees. How many full circles it is? 5.5*24*60/360 = 5.5*4 = 22. Each full circle contains 2 right angles (90 and 270). So answer is 22*2 = 44.

Calm down, we first must convert time to angle, two different units. The hour hand completes one full revolution each 12 hours (considering a 12 clock). So, theta_h = 360 x h/12, where theta_h is the angle that the hour hand makes with 12 hs mark and h is the number of hours since 0hs. So at 0 hs, the angle is 0, and at 12, the angle is 360 = 0. Since h = m/60, where m is the number of minutes since 0h, we have: theta_h = 360 x m/60 / 12 = 360 m / 12 x 60 = 0.5m Now, given the number of minutes since 0h, we can tell the angle of hour hand using theta_h. The minute hand angle, theta_m is: theta_m = 360 x m / 60 = 6m So the difference between theta_h and theta_m is |theta_h - theta_m| = 5.5m. Now given the minutes since 0h, we can tell the angle between hour and minute hand. Now, how many minutes we need to make |theta_h - theta_m| = 5.5m = 90? About 16.36. Since we have 12 x 60 minutes in a day, we have 12 x 60 / 16.36 gaps that satisfy 90 degrees, which is 44.