Here's the "definition" of night, if it helps:
According to the IRS, "A child is treated as living with a parent for a night if the child sleeps:
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At that parent's home, whether or not the parent is present, or
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In the company of the parent, when the child doesn’t sleep at a parent's home (for example, the parent and child are on vacation together)."
As pointed out by macuser_22, The IRS in Publication 17 also says:
"Parent works at night.
If, due to a parent's nighttime work schedule, a child lives for a greater number of days, but not nights, with the parent who works at night, that parent is treated as the custodial parent. On a school day, the child is treated as living at the primary residence registered with the school."
This may change things. Other rules to consider:
A non-custodial parent can never claim the Earned Income Credit, Head of Household filing status or the daycare credit, based on that child, even when the custodial parent has released the exemption to him. You'll have to determine if you are the custodial parent according to the references stated above.
Another way that you'll be able to claim the children is to fill out Form 8332 and obtain a release of claim for exemption from the other parent.
You will need the child's name, the tax year, your Social Security number, then the other parent's signature and date. If you prefer to get release of claim to your child's exemption for more than one tax year, enter the same information in part two rather than part one.
Once complete, file it with your tax return. If they release claim for multiple tax years, you only need to fill out the form once and attach copies of the original to your return each year.
ref: https://www.irs.gov/publications/p596/ch02.html
"A child will be treated as the qualifying child of his or her noncustodial parent (for purposes of claiming an exemption and the child tax credit, but not for the EIC) if all of the following apply:
1. The parents:
a. Are divorced or legally separated under a decree of divorce or separate maintenance,
b. Are separated under a written separation agreement, or
c. Lived apart at all times during the last 6 months of the tax year, whether or not they are or were married.
2. The child received over half of his or her support for the year from the parents.
3. The child is in the custody of one or both parents for more than half of the tax year.
4. Either of the following statements is true.
a. The custodial parent signs Form 8332 or a substantially similar statement that he or she will not claim the child as a dependent for the year, and the noncustodial parent attaches the form or statement to his or her return. If the divorce decree or separation agreement went into effect after 1984 and before 2009, the noncustodial parent may be able to attach certain pages from the decree or agreement instead of Form 8332.
A pre-1985 decree of divorce or separate maintenance or written separation agreement that applies to the tax year provides that the noncustodial parent can claim the child as a dependent, and the noncustodial parent provides at least $600 for support of the child during the tax year."
