How to connect more than 10 sensors to Arduino uno r3
You can easily increase number of I/O pins using an expander, e.g. PCF8574. 
The expander can be connect using your SCL/SDA pins located next to GND/AREF pins. (Here you can find detailed photo: http://forum.arduino.cc/index.php?topic=84190.0) You can connect up to 8 such expanders to extend the number of your digital pins to 64. Once you connect it, here is a code snippet of how to use it:

#include &lt;PCF8574.h&gt; //library available at: https://github.com/skywodd/pcf8574_arduino_library

void setup() {
  PCF8574 sensors;
  sensors.begin(0x20); //0x20 means that you connected A0, A1, A2 pins of PCF8574 to the ground
  sensors.pinMode(0, INPUT_PULLUP);
  sensors.pinMode(1, INPUT_PULLUP);
/*...*/
}
void loop() {      

  byte sensorValue = sensors.digitalRead(0); //reads digital value of 1st sensor connected to pin 0 of PCF8574 expander
  /*...*/
}


You can read more with this tutorial (note: they are using wire interface, so the code will differ. Choose the one you prefer).