A payment halved, a count halved, and the quantities that refused to follow

1

Both sides halved, and what obeys

Two offers of one shape sit either side of this card. The larger releases two hundred turns against a ten pound payment and the smaller releases a hundred against five, and neither asks for any play-through on the winnings. Price both at a turn costing 10p, which this card supposes supposed, at the published floor, which is where the range in this title’s own session data starts and a figure the reel is therefore confirmed to sell.

Priced that way, a hundred turns commit £10.00 of stake, and a published 96.71 per cent turns the £10.00 into an expected £9.67, and with no multiple written anywhere the £9.67 is the close. Two hundred turns commit £20.00 and close at £19.34. So the headline figure halves, the stake halves, the yield halves and the close halves, and anybody who stopped reading here would conclude that one arrangement is simply the other at half the size.

The first five rows are the halving doing as it is told. The last three are the whole subject of this card, and they are not a fault in the offer or a trick in the terms. They are what happens when one side of a multiplication is denominated in the grant and the other in the machine, and only one of the two was halved.

2

Every rate in the arrangement holds still

Before the asymmetry, the part a reader can rely on. Expectation on this site is E = B × (1 − W × (1 − RTP) ∕ g), and on both of these offers W is nought, so the bracket is one and the close is B on either size. B on the smaller is the £9.67 a hundred turns should produce rather than the £10.00 they stake, and B on the larger is £19.34; the weighting g never gets a chance to act, since there is no requirement for a slot to be weighed against a term of the grant rather than of the reel; and RTP carries the title’s one published figure in both columns.

Because the close is strictly proportional to the count, every per-pound and per-turn figure survives the halving untouched. Twenty turns a pound either way. £1.93 of expected winnings a pound either way. Of anything crossing the reel, 3.29 points stays behind, so a 10p turn gives back 9.671p on either offer. Break-even on this title is 30.4 times whichever grant is in play, being the reciprocal of the reel’s share and no business of the offer’s at all.

That matters because it rules out the obvious complaint. The smaller arrangement is not worse value per pound, not worse value per turn, and not subject to a harsher condition, there being no condition on either. A reader choosing between them on any rate at all is choosing between two identical things, and a reader choosing on magnitude is getting exactly what the headline says. The difference this card is for does not show up in any rate.

3

The ceiling stays put, so in grants it doubles

The most one turn of this title can produce is a win limit of 2,100 stakes the limit, off the session data, and no maximum win appears in the maker’s page for the title at all, so that figure is the game’s rather than the page’s. At 10p a turn it is £210.00. Notice what the figure is attached to: a stake. It is not attached to the grant, to the payment or to the count, so halving any of those three leaves it precisely where it was.

Set it against the thing that did halve and the arithmetic is quick. £210.00 against the £19.34 the larger grant should yield is 10.9 grants. The same £210.00 against the £9.67 the smaller one should yield is 21.7. So one turn of this reel can be worth twice as many of the smaller arrangements as of the larger, and nobody wrote that into either offer: it is a consequence of the ceiling being quoted in stakes while the grant is quoted in turns.

The same thing happens to every quantity the maker prints as a multiple of one stake. The largest money symbol the round can collect carries 2,000 times the stake off the maker’s own page, which is £200.00 at this price: 10.3 of the larger grants, 20.7 of the smaller. The gap between those two published figures is a hundred stakes, which is £10.00 here, and £10.00 reads very differently against a grant of £9.67 than against one of £19.34, being a whole grant in the first case and half of one in the second.

Six pence and eighty-two grants, in one row, off the same two titles. The published return decides the fourth column and moves it by sixpence; the published ceiling decides the fifth and moves it by more than eighty times what the grant is expected to be worth. Both figures are the maker’s and both are a property of the title, and a reader picking a name off a lobby tile has picked both of them at once.

4

The round is counted in turns and not in grants

The feature round is the other place the halving fails to arrive, and for the same reason in a different unit. Ten awarded turns come with three scatters, fifteen with four and twenty with five the map, from page and data alike. Those are counts of turns, fixed by the machine, and the grant has no say in them.

So the biggest of those awards is a fifth of a hundred-turn grant where it would have been a tenth of a grant of two hundred. Each fourth wild the round collects brings ten further turns with it, which is a twentieth of the larger grant and a tenth of the smaller, and the win multiplier steps to twice, then to three times, then to ten across the first three of those extensions. Getting to the top rung therefore takes twelve wilds inside the round and brings thirty turns with it, so such a round has lasted at least forty turns, a fifth of the larger grant and two fifths of the smaller.

Put those fractions in a line and the pattern of this whole card appears again. A hundred turns is a grant inside which one round is a bigger event, one retrigger is a bigger event, and a round climbing the ladder is a much bigger event, every time by exactly the factor of two the offer applied to itself. What the maker does not state is how often any of that happens: the round’s arrival rate stands nowhere on the maker’s pages and has no field in a session’s data, and the two rates review sites quote lie more than half again apart. So the fractions above are sizes and not chances, and this card prints no chance for them.

A single turn of this title may hand over 21.7 of these hundred-turn grants and 10.9 of the two-hundred-turn ones, the ceiling being quoted in stakes while a grant is quoted in turns

5

Where the doubling is worth precisely nothing

Here is the part that keeps the previous two entries honest, and it is the reason this card does not end by calling the smaller offer worse. The one frequency this title publishes about itself is for the ceiling: once every 3,880,481 turns the hit rate, out of the same data. That rate is counted per turn, so halving the count halves the number of draws against it, from two hundred to one hundred.

Multiply the two movements together and they cancel exactly. The ceiling measured in grants doubled; the draws against it halved; so the ceiling’s contribution to the expected winnings is the same fraction of either grant. Worked out, the contribution is a little over half a penny on the smaller grant and a little over a penny on the larger, and in both cases it is one part in 1,787 of what the grant should yield. The most advertised thing this machine can do pays for a 1,787th of either arrangement, and halving the arrangement did not change that by a hundredth of a penny.

That is what the halving actually did, stated properly. It did not make the smaller grant worse in expectation, because expectation is linear in the count and nothing in the machine’s own figures is nonlinear. It made the smaller grant a coarser instrument for measuring the same machine: the same events, the same rates, the same average per turn, with each event counting for twice as much of the total and arriving half as often. A reader who wants a steadier result wants more turns, and a reader who wants the result to be mostly decided by one event wants fewer, and the two wishes are the same wish in opposite directions.

What nobody can supply is the figure that would turn that into a number. How far the outcome of a hundred turns strays from £9.67 belongs to the reel, and the maker lists no title with a spread against its name. Not one of its game pages carries the word, nothing a session hands over has anywhere to keep it, and so the only honest arithmetic left is the kind done above: a ceiling, the one rate behind it, and the counts a grant supplies.

6

One halving, run across three published ceilings

The family does not publish one ceiling, it publishes several, and that gives the halving somewhere to be tested. Three titles are enough. The win limit is 2,100 stakes on Big Bass Bonanza, 5,000 on Big Bass Splash and 10,000 on Big Bass Day at the Races three limits, each off its own data, which at 10p a turn is £210.00, £500.00 and £1,000.00. The first two publish the same 96.71 per cent and the third publishes 96.07, so the grants they yield are £9.67, £9.67 and £9.61 at a hundred turns and twice those at two hundred.

Read the columns and the ratio between the two rows never moves: 21.7 against 10.9, 51.7 against 25.9, 104.1 against 52.0. Whatever ceiling a title publishes, halving the grant doubles that ceiling measured in grants, which is what this card was written to establish and the reason the drawing has no total on it. The three columns are three accounts of one halving, not three parts of anything, and adding them would state a quantity of nothing.

Read the rows instead and the spread is where the title choice lives: 41.1 grants across the three ceilings at the larger size, 82.4 at the smaller. So the smaller grant does not only magnify the ceiling, it magnifies the difference between ceilings, and a reader who takes a hundred turns rather than two hundred has made the name on the tile matter twice as much as it did.

7

Four minutes ten, and what a licence has to do with it

The clock halves as cleanly as the money. On a licensed build nothing may follow a cycle inside 2.5 seconds, each one takes its own commitment, and the build may not run itself the pace floor, RTS 14D, so a hundred turns cannot be spent inside 250 seconds. Four minutes and ten seconds is the smaller grant at the outside pace, against eight minutes twenty for the larger, and no hour of the same build can hold more than 1,440 turns.

That caps what either arrangement can be worth per hour of a reader’s time, which is the only sense in which these grants have an hourly rate at all. A hundred turns yielding £9.67 in 250 seconds is a great deal per hour and lasts four minutes; the £9.67 does not continue arriving. What continues, if a reader stays at the machine with the £9.67 in hand, is the reel’s 3.29 points on everything further that crosses it, and the smaller grant buys a shorter protection against that than the larger one does for the plain reason that it is smaller.

8

What the smaller arrangement actually is

Collect it. A hundred turns at the ten pence this title is sold from commit £10.00 and should hand back £9.67, which is the close as well, nothing being asked of it. Twenty turns a pound and £1.93 a pound are the same figures the larger offer gives, so no rate has moved. The grant takes four minutes and ten seconds at the briskest a licensed build runs. Discharged on a title published 0.64 of a point lower the same hundred turns are worth £9.61, so the choice of title is worth sixpence. And the most one turn may hand over is £210.00 on this title either way, which is 21.7 grants of this size and 10.9 of the larger.

So the answer to what halving both sides changes is not nothing and not very much either, and it is in a column most readers never look at. Everything the offer controls halved. Everything the machine controls stayed exactly as it was. The result is the same average per turn arriving in a package half the size, which makes every one of the machine’s own fixed quantities twice as large relative to the package and makes the package half as likely to meet any of them. The two effects multiply out to one, so the expectation is untouched and the shape of the outcome is not.

There is a practical reading of that, and it is the only advice-shaped sentence this card will produce. If the figure a reader cares about is the average, the two offers are the same offer and the smaller one is simply smaller. If the figure a reader cares about is what the outcome actually looks like, the smaller offer is the coarser of the two instruments, and the quantity that would say by how much is published nowhere by anybody. The question invites a larger answer than that, and the published figures will not carry one.

One figure deliberately absent, for the record. Nothing above says how likely the feature round is inside a hundred turns, or inside two hundred. No arrival rate for it is printed by the maker, nothing in a session’s data has room for one, and the pair of figures review sites carry sit far enough apart that an average of them would be a third invention. Every fraction in entry four is a size, every ratio in entries three and six is a ratio of published figures, and the one rate used anywhere on this card is the one the game states for its own ceiling.