On April 25, there will be no shadow at noon when the sun will be exactly at zenith. This will be demonstrated at 12:17 hrs at Jawaharlal Nehru Planetarium. It requires no special gadget to see this effect; any pillar will have no shadow.
On June 22, the sun will be at the northernmost point - this is called summer solstice or Dakshinayana. The sun, then retraces the path towards south. Again on August 19, the sun will be exactly at zenith here. Thus the shadow at noon can be zero only on two specific days in a year for places between the Equator and the Tropic of Cancer. The dates vary from place to place.
The attached list gives the dates for the district head quarters of Karnataka. The timings range from 11:55 to 12:25 from place to place. The exact time can be determined by watching the disappearance of the shadow of a glass tumbler.