While there does not appear to be any reference on Form 1040 nor in the IRS instructions requiring a Social Security Number (SSN) be entered for the qualifying child when a return's filing status is Head of Household (HoH), the IRS does have a rejection code that will reject any return electronically filed that does not contain an SSN for a qualifying child for HoH purposes. Due to this rejection code, the only way to file a return without an SSN for the qualifying child would be to paper file the return.