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If I were you I'd be prepared for an early morning. Other attractions I have visited offering free passes on a first come first served basis usually tend to attract massive queues hours before the opening of the ticket office. This also send to be true for the Washington Monument:
NOTE FOR SAME DAY TICKETS TO THE WASHINGTON MONUMENT:
If you plan on getting same-day tickets to the Washington Monument during Spring/Summer months, expect an early morning.
Ticket handouts begin at 8:30 am, but the line starts to form as early as 7 am
Tickets are given out on a first come, first served basis, so morning times tend to go fast.
Each person can obtain up to 6 tickets.
Tickets can be fully distributed by 9:30 am.
Visitor’s Tip:
If you don’t get same day tickets, you may still be in luck.
School groups visiting between March-June often order too many tickets in advance. Since they cannot use them all, many groups will hand out tickets to guests waiting around the base of the memorial.
There is no way to time this so fate has to be on your side, but if its a nice day and you see a lot of prepubescent middle-schoolers in matching t-shirts, it may be worth hanging out to see!