Why do newspaper headlines use strange syntax rules?

Solution 1:

This is ellipsis, but more importantly, English headlines follow special conventions that are, by and large, consistent across publications. Headlines have evolved to maximize information output and minimize space, because this has been optimal for newspapers (until the Internet age, at least — but now the conventions are ingrained into the world of journalism, needed or not).

This headline style guide covers the conventions in great detail.

Relevant quotes:

In many headlines, as with the example immediately above (…loophole [is] ‘too big’), the verb “to be” is not necessary. It can be used, but in most cases should be avoided.

Present tense, please: Use present tense for immediate past information, past tense for past perfect, and future tense for coming events.

Avoid the use of the articles a, an and the unless they are needed for clarity. (Otherwise, their use generally is considered padding.)

The comma, in addition to its normal use, can take on the work of the word “and.”

Solution 2:

That is called ellipsis, omitting words that can be inferred. Journalists use such strong ellipsis because they want compact headlines. The present tense is used here to make a story seem more "actual", more lively: that is called the historic present. Besides this liveliness, the fact that it is often shorter is practical for journalists. The rules for ellipsis are, as far as I know, no more specific than that, in a headline, anything that can be easily inferred may be left out. Sometimes journalists leave out too much, confusing the reader.

Solution 3:

The reason?

Not Much Room, Short Implied Words Removed

Another name for this is "telescopic writing" which is also used on street signs.

The reason is to use less space on the page. I wouldn't call these 'rules' but rather common patterns of writing.

Some other patterns:

• reduced use of punctuation
• noun modifiers: "Poker advocate: Estimated $100M-$500M in refunds for online players" instead of "Advocate of poker...", or shorter country names rather than adjectives e.g. "Spain PM" instead of "Spanish PM"
• giving a name or noun to introduce, then a detail after a colon. Officials: No immediate al-Qaeda threat to U.S. (instead of saying "Officials say...")
• dropping helper verbs: "Weak dollar helping U.S. profits" instead of "...is helping..."
• removing verbs altogether "Google's animated tribute to dance pioneer Martha Graham" instead of "Google gives animated tribute..."