The Cards That Count for Nothing... Or Do They?

Las Vegas, 1963. At a computing conference, an engineer named Harvey Dubner handed out index cards describing a way to beat blackjack so simple that, by his wife's account, the room nearly rioted to grab a copy. His idea — later sharpened by Julian Braun and Stanford Wong, and published in Edward Thorp's Beat the Dealer — is still the most widely used card-counting system in the world. It is called Hi-Lo, and its genius is compression: it folds a shoe of hundreds of cards into a single number.

That number answers one question — is what's left in the shoe richer in big cards than a fresh deck would be? Big cards (tens and aces) help you: they make blackjacks, which pay 3:2, and they bust the dealer on stiff hands. So Hi-Lo gives every card a tag:

2 3 4 5 6 → +1 · 7 8 9 → 0 · 10 J Q K A → −1

You keep a running count: start at zero and add each card's tag as it appears. Two definitions, so nothing stays mysterious:

Running count (RC). The tally so far. A positive RC means more low cards than high cards have already left, so the rest of the shoe is rich in big cards — good for you.
True count (TC). The running count divided by the decks still unplayed: TC = RC ÷ decks remaining. A running count of +6 means a lot with one deck to go and very little with five, so we normalize before trusting it.

Hi-Lo is balanced: a single deck holds twenty low cards (+1 each), twenty high cards (−1 each), and twelve "neutral" 7-8-9s (0 each). Twenty up, twenty down, the rest blank — it all cancels to zero, so counting down a full deck lands you right back where you began.

Here is the snag this puzzle is built on. The 7, 8 and 9 are scored nothing. Yet those cards plainly do things — they lift a 12 to a 19, they let a dealer crawl off a stiff. Throwing away a quarter of the deck feels wasteful; surely a sharper count would use them. So we will test three ways to draw the line between "low" and "high":

Option a — Standard Hi-Lo. 2–6 = +1, 7-8-9 = 0, tens and aces = −1.
Option b — Low-stretch. Fold the middle cards in with the lows: 2–9 = +1, tens and aces = −1.
Option c — High-stretch. Fold the middle cards in with the highs: 2–6 = +1, 7 through ace = −1.

The challenge. Which of the three best flags the shoes where your edge is highest? And, more generally: as you slide the dividing line anywhere from the 2 up to the ace, where does the count work best — and can any hard line beat the standard one that leaves three cards blank?

Assumptions. Six-deck shoe (we'll also vary it from one to twenty); blackjack pays 3:2; doubling and splitting allowed; dealer stands on soft 17; you raise your bet when the count says the shoe is favorable; judge a session of 64 hands, reshuffle included.

Interactive Supplement

Cards That Count For Nothing

Explore this puzzle visually with an interactive diagram — drag sliders, watch the geometry update in real time, and build intuition before you solve.

Open interactive →

💡 Hint

A counting tag is worth something only if it matches how much your edge moves when that card leaves the shoe — not how often the card shows up in a winning hand. Look up the "effect of removal" for the 7, the 8 and the 9. One of those three is secretly a high card, and forcing all three onto either side adds more noise than signal.

Solution

The trap is Option b — and it's a good one. "Use more of the deck" sounds like strictly more information, and more information can't hurt… can it? It can. The worth of a counting tag has almost nothing to do with how often a card turns up in a winning hand. It depends on one thing: how much your advantage moves when that card is removed from the shoe. Counters call this the effect of removal.

Peter Griffin measured it for every rank in The Theory of Blackjack. Read each figure as "the change in the player's edge, in percent, when one such card is taken out of a single deck." Positive means removing it helps you:

Card	Effect of removal	what it means
2	+0.38%
3	+0.44%
4	+0.55%
5	+0.69%	most valuable to remove — the dealer's best friend
6	+0.46%
7	+0.28%	faintly good to lose — a touch low-like
8	−0.00%	almost exactly nothing — sits on the fence post
9	−0.18%	faintly bad to lose — a small high card in disguise
10 J Q K	−0.51%
A	−0.61%	losing it collapses your edge — no more 3:2 naturals

The pattern is the whole story. The 5 is the dealer's best friend — drop it and you gain the most, because it's the card that rescues stiff hands. Tens and aces are the mirror image. And the three "neutral" cards aren't a tidy little group at all: the 7 is faintly low-like, the 8 is essentially nothing, and the 9 quietly behaves like a high card. They straddle the fence, and the 8 sits on the post.

To grade a whole system in one number, ask how well its tags line up with those true effects. That measure is the betting correlation — a correlation coefficient between each rank's tag and its effect of removal. Picture plotting all thirteen ranks, "tag" along the bottom and "true effect" up the side: if the dots hug a rising straight line the correlation is near 1.0 (a flawless detector); if they scatter it slides toward 0. Running the numbers:

Option a — Standard Hi-Lo: 0.968
Option b — low-stretch: 0.891
Option c — high-stretch: 0.855

Option a wins, and the reason is signal versus noise. When Option b scores the 7-8-9 as +1, it gets exactly one thing right (the 7 really is a little low-like) and one thing wrong (a +1 on the 9 points the wrong way), with the 8 a coin flip. The sliver it gains is swamped by the cost: it has bolted three fresh ±1 swings onto the count that mostly don't track your edge. A correlation divides useful alignment by the total size of the wiggle — and Option b's wiggle grew far faster than its usefulness. Option c is worse still: it brands the genuinely low-like 7 a high card. Hi-Lo's blanks are not wasted; they are the system declining to bet on cards it cannot read, which is precisely what keeps the signal sharp. Anyone who has watched one junk variable wreck a regression will recognize the move.

The deeper twist. Drop the blanks entirely and hunt for the single best place to cut the deck into "low +1" and "high −1." Slide that line across every rank and the betting correlation traces a clean hill: it climbs out of the low cards, crests at 0.928 with the line just after the 8, then tumbles as you drag it into the tens. Option c (line after the 6) and Option b (line after the 9) are the two shoulders of that hill — both below the summit. And the summit itself, the best hard line you can draw, still loses to standard Hi-Lo's 0.968. Choosing to leave the three fence-sitters blank beats every place you could have forced them. Doing less wins.

Do more decks rescue Option b or c? No. The effect-of-removal pattern is nearly identical whether you play one deck or six — the two patterns correlate above 0.999 — so the betting correlations, and the order a > b > c, barely twitch from one deck to twenty. What the deck count does change is the size of the prize: a deeper shoe shifts less with every card removed, so favorable moments grow rarer and milder. The edge shrinks with more decks; the winner never changes. The supplement lets you drag both the dividing line and the deck count and watch exactly this.

One last wink from the real world. Counting only the 7 (and nothing else extra) is a genuine, respected system — the unbalanced Knock-Out count — and it nudges the betting correlation a hair above Hi-Lo, precisely because the 7 is the one fence-sitter worth catching. But scoring all three middle cards, as Options b and c do, overreaches. Ken Uston once treated the 7 as a high card and the 2 as neutral; the "Green Fountain" systems tried scoring the 7 and 9; the standard verdict is that such tweaks are mistakes that roughly cancel. The middle of the deck is treacherous. Sometimes the sharpest thing a counter can do is count it for nothing.

Answer: Option a — standard Hi-Lo.